UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO
POSGRADO EN CIENCIAS MATEMÁTICAS
FUNCIONES CARDINALES Y ESPACIOS DE FUNCIONES
CONTINUAS
TESIS
QUE PARA OPTAR POR EL GRADO DE:
DOCTOR EN CIENCIAS
PRESENTA:
REYNALDO ROJAS HERNÁNDEZ
DIRECTOR DE TESIS
DR. ÁNGEL TAMARIZ MASCARÚA, FAC. DE CIENCIAS, UNAM
MIEMBROS DEL COMITÉ TUTOR
DR. FERNANDO HERNÁNDEZ HERNÁNDEZ, FAC. DE CIENCIAS
FÍSICO MATEMÁTICAS, UMSNH
DRA. ISABEL PUGA ESPINOSA, FAC. DE CIENCIAS, UNAM
MÉXICO, D.F., SEPTIEMBRE 2013
 
UNAM – Dirección General de Bibliotecas 
Tesis Digitales 
Restricciones de uso 
  
DERECHOS RESERVADOS © 
PROHIBIDA SU REPRODUCCIÓN TOTAL O PARCIAL 
  
Todo el material contenido en esta tesis esta protegido por la Ley Federal 
del Derecho de Autor (LFDA) de los Estados Unidos Mexicanos (México). 
El uso de imágenes, fragmentos de videos, y demás material que sea 
objeto de protección de los derechos de autor, será exclusivamente para 
fines educativos e informativos y deberá citar la fuente donde la obtuvo 
mencionando el autor o autores. Cualquier uso distinto como el lucro, 
reproducción, edición o modificación, será perseguido y sancionado por el 
respectivo titular de los Derechos de Autor. 
 
  
 
Este trabajo de tesis doctoral recibió el apoyo económico de CONACyT,
beca 213711, y de PAPIIT-DGAPA-UNAM IN 115312.
Introducción
La obtención del primer cúmulo de resultados en Cp-teoŕıa se remonta a
los años 60 y 70 del siglo pasado. En su mayoŕıa pertenecen a especialistas en
Análisis Funcional y sirven a propósitos espećıficos en sus áreas, sin embargo,
los topólogos pronto se dieron cuenta de que un buen número de teoremas del
Análisis Funcional conteńıan hechos topológicos no triviales muy interesantes.
Como consecuencia de ello, un estudio detallado y sistemático de la esencia
topológica de los avances mencionados se llevó a cabo por A.V. Arhangel’skii
y su escuela. En efecto, algunos avances fundamentales fueron obtenidos por
A.V. Arhangel’skii, S.P. Gul’ko, E.G. Pytkeev, O.G. Okunev, D.P. Baturov,
E.A. Reznichenko, V.V. Tkachuk y otros autores. Como consecuencia, una
buena cantidad de resultados heterogéneos pudo unificarse para formar una
teoŕıa coherente y hermosa, que no sólo ofrece un poderoso apoyo para las
necesidades topológicas de otras áreas de las matemáticas, como el Análisis
Funcional, la Teoŕıa Descriptiva de Conjuntos y el Álgebra Topológica, sino
que además tiene un impresionante potencial de desarrollo interno.
La noción de espacio D fue introducida por E.K. van Douwen [25]. Estos
espacios han atráıdo gran atención en los últimos años, sin embargo, varios
hechos fundamentales acerca de estos espacios permanecen desconocidos. Un
hecho muy interesante, que fue observado en el art́ıculo citado, es el siguiente:
“Hasta ahora no es conocido ningún ejemplo satisfactorio de un espacio que
no sea D, donde por ejemplo satisfactorio nos referimos a un ejemplo con
una propiedad tipo cubierta al menos tan fuerte como metacompacidad o
submetacompacidad”. Probablemente la mejor pregunta abierta conocida
acerca de estos espacios es la siguiente: ¿Es cierto que todo espacio Lindelöf
regular es un espacio D?
Los espacios D de E.K. van Douwen han sido objeto de estudio intensivo
en los últimos años. Ellos han sido estudiados en casi todos los contextos y
la Cp-teoŕıa no ha sido una excepción. La noción de monoliticidad monótona
introducida por V.V. Tkachuk ha surgido durante el estudio de la propiedad
D en los espacios de funciones.
El propósito de este trabajo es describir el estado actual de la investigación
en espacios monotónamente monoĺıticos, en general y en espacios de funciones,
y presentar nuestros resultados en este tópico.
v
vi INTRODUCCIÓN
En la primera parte de nuestro trabajo se introduce la topoloǵıa de la
convergencia puntual, describimos algunas construcciones básicas y presenta-
mos algunos resultados conocidos que nos serán útiles. Una referencia muy
reciente y completa en Cp-teoŕıa, que recomendamos, es el libro de V.V.
Tkachuk [70]. Después de esto, se describe de forma breve el estado actual
de la investigación en espacios D. Sin embargo, recomendamos el art́ıculo de
G. Gruenhage [36] que proporciona una revisión completa de lo que sabemos y
no sabemos sobre espacios D. Posteriormente, se incluye una sección sobre la
propiedad D en espacios metalindelöf, donde incluimos un resultado original
junto con su correspondiente demostración. En el resto de la primera parte
se describe el estado actual de la investigación en espacios monotónamente
monoĺıticos, incluyendo resultados en espacios monotónamente monoĺıticos,
espacios débilmente monótonamente monoĺıticos, espacios monótonamente
κ-monoĺıticos, espacios fuertemente monótonamente monoĺıticos y espacios
con la propiedad de Collins-Roscoe. Todos estos resultados muestran que la
monoliticidad monótona es una propiedad útil e interesante.
En la segunda parte, presentamos nuestros aportes en este tópico, dando
pruebas detalladas de todos los resultados. Introducimos los espacios monóto-
namente estables. Se introduce la estabilidad monótona y se muestra que ex-
iste una dualidad entre la estabilidad monótona y la monoliticidad monótona
en los espacios de funciones. Presentamos algunos resultados sobre monoliti-
cidad monótona en espacios de funciones sobre productos y Σ-productos. Por
otra parte, se da una generalización de un teorema de O.G. Okunev acerca
de los espacios de funciones sobre conjuntos conulos. También, probamos
algunos resultados sobre espacios monotónamnete κ-monoĺıticos e introduci-
mos la noción de espacio montónamente <κ-monoĺıtico. Algunos resultados
sobre espacios monótonamente monoĺıticos se extienden para espacios mono-
tonamente <κ-monoĺıticos y se utilizan para encontrar nuevos espacios con la
propiedad D. Al final de esta parte, incluimos algunos resultados acerca de
la monoliticidad monótona y la propiedad Collins-Roscoe en Σs-productos,
que generalizan algunos resultados conocidos. Varias preguntas abiertas se
responden usando estos resultados.
Por último, se presenta una lista de problemas sobre la propiedad D, la
monoliticidad monótona y los espacios de funciones, que permanecen abiertos
(para el conocimiento del autor). Se describe cada problema y se proporcionan
las referencias correspondientes.
Contents
Introduction v
Chapter I. Preliminaries 1
1. Terminology and Notation 1
2. Function spaces 2
3. Monolithic and stable spaces in Cp-duality 4
4. D-property 6
5. D-property and metalindelöf spaces 10
6. D-property in function spaces 13
7. Monotone monolithicity 15
8. Weakly monotone monolithic spaces 17
9. Monotone κ-monolithicity 18
10. The Collins-Roscoe property 20
11. Monotone monolithicity in compact spaces 22
12. The Collins-Roscoe property in function spaces 26
13. Strong monotone monolithicity 28
Chapter II. Monotone monolithicity 31
1. Monotone monolithicity of Cp(Cp(X)) 31
2. When is Cp(X) monotonically monolithic? 35
3. Monotone monolithicity of Cp(X) when X is a Σ-product 40
4. Monotone monolithicity of Cp(X) and cozero sets 44
5. Monotone κ-monolithicity and function spaces 47
6. Monotone <κ-monolithicity 52
7. Monotone monolithicity and Σs-products 59
Open problems and questions 65
Bibliography 71
vii
CHAPTER I
Preliminaries
1. Terminology and Notation
In notation and terminology we follow [29]. Every space in this work is a
Tychonoff space with more than one point. The letters α, β and γ represent
ordinal numbers and the letters λ and κ represent infinite cardinal numbers;
ω is the first infinite cardinal, ω1 is the first non-countable cardinal. The
ordinal number α with its order topology will be symbolized by [0, α). We
denote the space [0, α + 1) as [0, α]. Also, we denote the ordinal number α
with its discrete topology simply as α. For a subset A of a topological space
X, clX(A) is the closure of A in X, and intX(A) is the interior of A in X.
If there is no possibility of confusion, we will write simply cl(A) and int(A)
instead of clX(A) and intX(A).
Given a cardinal number, a subset A of a space X is a set of type Gκ if it
is the intersection of at most κ open sets in X.
By βX we denote the Stone-Čech compactification of a space X. Given
an infinite cardinal κ, the Hewitt κ-extension of X is the space υκX = {x ∈
βX : if F is a Gκ-subset of βX and x ∈ F then F ∩ X 6= ∅}. Besides, the
Hewitt extension υωX of the space X is canonically homeomorphic to υX.
Let E and N be two families of subsets of X. We say that N is a network
for X modulo E if for each E ∈ E and each open subset U of X with E ⊂ U ,
there is N ∈ N satisfying E ⊂ N ⊂ U .
A space X is a Lindelöf Σ-space if X possesses both a cover K constituted
by compact subsets of X and a countable network N for X modulo K.
A family N of subsets of X is a network in X if it is a network for X
modulo {{x} : x ∈ X}. A space X is cosmic if it has a countable network.
A function f : X → Y is called condensation if it is a continuous bijection;
in this case we say that X condenses onto Y . If X condenses onto a subspace
of Y we say that X condenses into Y .
A subspace Y of a space X is C-embedded in X if each real-valued con-
tinuous function can be extended to a real-valued continuous function on all
of X.
The Lindelöf number l(X) of a space X is the smallest infinite cardinal κ
such that any open cover of X contains a subcover of cardinality at most κ.
The extent e(X) of X is the smallest infinite cardinal κ such that the
cardinality of every closed discrete subspace of X does not exceed κ.
The density d(X) of X is the minimal cardinality of a dense set in X.
1
2 I. PRELIMINARIES
iw(X) denotes the minimal cardinality of a space Y onto which X can be
condensed. The cardinal invariant iw(X) is called the i-weight of X.
The network weight nw(X) is the minimal cardinality of a network in X.
The tightness t(X) of a space X is the smallest infinite cardinal κ such
that for any set A ⊂ X and any point x ∈ cl(A), there exists a set B ⊂ A
with |B| ≤ κ and x ∈ cl(B).
For a cardinal invariant φ we define the new cardinal invariant φ∗ as
follows: φ∗(X) = sup{φ(Xn) : n ∈ ω \ {0}}.
Suppose that η = {Xt : t ∈ T} is a family of topological spaces, X =
∏
{Xt : t ∈ T} is the topological product of the family η, and x∗ is a point
in X. Then the Σ-product of η with basic point x∗ is the subspace of X
consisting of all points x ∈ X such that only countably many coordinates
x(α) of x are distinct from the corresponding coordinates x∗(α) of x∗. This
subspace is denoted by Σ{Xt : t ∈ T} or by Ση. Sometimes Ση will be called
the Σ-product of η at x∗. Similarly, the σ-product of η with basic point x∗ is
the subspace of X consisting of all points x ∈ X such that only finitely many
coordinates x(a) of x are distinct from the corresponding coordinates x∗(a)
of x∗. This subspace is denoted by σ{Xt : t ∈ T} or by ση. Sometimes ση
will be called the σ-product of η at x∗. Note that the basic point is usually
not shown in the notation.
A collection U of subsets of X is said to be point-countable if each point
x ∈ X is an element of at most countably many members of U .
ω1 is a caliber of a space X if every point-countable family of non-empty
open subsets of X is countable.
For a space X, CL(X) denotes the family of all non-empty closed subsets
of X, and K(X) denotes the family of all non-empty compact subsets of X.
For those concepts and notations which appear without definition, consult
[29] and [70].
2. Function spaces
The creation of the Cp-theory or theory of function spaces endowed with
the topology of pointwise convergence must be attributed to Alexander Vla-
dimirovich Arhangel’skii. A.V. Arhangel’skii was the first to understand the
need to unify and classify a bulk of heterogeneous results from topological al-
gebra, functional analysis and general topology. He was also the first to obtain
crucial results that made this unification possible and the first to formulate
a critical mass of open problems which showed this theory’s huge potential
for development. Later, many mathematicians worked hard to contribute to
Cp-theory giving it the elegance and beauty that it boasts today.
In this section we introduce the topology of pointwise convergence in
spaces of continuous functions. Also, we present some basic constructions
and results in function spaces that are well known and will be useful later.
All the proofs can be found in [3].
2. FUNCTION SPACES 3
For all spaces X and Y we denote by C(X, Y ) the set of all continuous
functions from X to Y . We set C(X) = C(X,R). By Cp(X, Y ) we denote
the set C(X, Y ) endowed with the topology of pointwise convergence. The
standard base of the space Cp(X, Y ) consists of all the sets of the form:
[x1, . . . , xk : U1, . . . , Uk] = {f ∈ C(X, Y ) : f(xi) ∈ Ui, i = 1, . . . , k},
where x1, . . . , xk are elements of X, U1, . . . , Uk are open sets in Y and k ∈ ω.
Notice that the topology of pointwise convergence in C(X, Y ) is the topology
induced from the topological product Y X .
We denote by Cp(X) the space Cp(X, Y ). Clearly, the family of all the
sets of the form:
[f, x1, . . . , xk, ǫ] = {g ∈ C(X) : |g(xi)− f(xi)| < ǫ, i = 1, . . . , k},
where f ∈ C(X), x1, . . . , xk are elements of X and ǫ is a positive real number,
is a base for the space Cp(X).
Given a space X let Cp,0(X) = X and Cp,n+1(X) = Cp(Cp,n(X)) for all
n ∈ ω, i.e., Cp,n(X) is the nth iterated function space of X.
Let Y be a subspace of X. By πY we denote the function from Cp(X) to
Cp(Y ) which restricts each element in Cp(X) to Y ; that is, πY (f) = f ↾ Y .
The function πY is called the restriction function, and between its properties
we have the following:
Proposition 2.1. For every subspace Y of X the following hold:
(1) the function πY is continuous and cl(πY (Cp(X))) = Cp(Y );
(2) if Y is closed in X, then πY is an open function from Cp(X) onto
the subspace πY (Cp(X)) of Cp(Y );
(3) if Y is compact, then πY (Cp(X)) = Cp(Y );
(4) if X is normal and Y is closed in X, then πY (Cp(X)) = Cp(Y );
(5) if Y is everywhere dense in X, then πY : Cp(X) → πY (Cp(X)) is a
condensation.
Let f : X → Y be a function between the setsX and Y . The dual function
f ∗ : RY → R
X is defined as follows: if g ∈ R
Y , then f ∗(g) = g ◦ f . The
function f ∗ is called the dual function and it has the the following properties:
Proposition 2.2. Let f be a function from a set X to a set Y , then:
(1) f ∗ is a continuous function;
(2) If f(X) = Y , then f ∗ : RY → R
X is a homeomorphism from R
Y
onto the closed subspace f ∗(RY ) of RX .
Proposition 2.3. Let f : X → Y a function from X onto Y , then:
(1) f is continuous if and only if f ∗(Cp(Y )) ⊂ Cp(X);
(2) if f is a quotient map, then f ∗(Cp(Y )) is a closed subspace of Cp(X);
(3) f is a condensation if and only if f ∗(Cp(Y )) is everywhere dense in
Cp(X);
(4) f is a homeomorphism if and only if f ∗(Cp(Y )) = Cp(X).
4 I. PRELIMINARIES
Let f : X → Y be a map from a topological space X onto a set Y . Then
the strongest of all completely regular topologies on Y relative to which f is
continuous is called the R-quotient topology on the set Y . A function from a
space X onto a space Y is called an R-quotient function, if the topology on
Y coincides with the R-quotient topology generated by f .
The statement in 2) Proposition 2.3 has no immediate converse, however,
we have the following result.
Proposition 2.4. A function f from a space X onto a space Y is an
R-quotient map if and only if f ∗(Cp(Y )) is a closed subspace of Cp(X).
Suppose we are given a set X and a family F ⊂ R
X . The function
∆F : X → R
F given by [∆F(x)](f) = f(x) for each x ∈ X and f ∈ F is
called the diagonal of the family F . Clearly, if X is a topological space and
each f ∈ F is a continuous function, then ∆F is a continuous function.
Let ψ = ∆Cp(X). The following results are well known.
Proposition 2.5. For any space X:
(1) ψ(X) ⊂ Cp(Cp(X));
(2) ψ embeds X in Cp(Cp(X)) as a closed subspace.
The following results show the Cp-duality between some cardinal func-
tions.
Theorem 2.6. For any space X:
(1) nw(X) = nw(Cp(X));
(2) d(X) = iw(Cp(X));
(3) iw(X) = d(Cp(X)).
3. Monolithic and stable spaces in Cp-duality
The concepts of monolithic space and stable space were introduced by
A.V. Arhangel’skii in [4]. Given an infinite cardinal number κ, a space X is
called κ-monolithic if nw(cl(A)) ≤ κ for every subset A of X of cardinality
at most κ. In particular, X is ω-monolithic if the closure of every countable
set in X is a space with countable network.
A space X is called monolithic if it is κ-monolithic for every cardinal κ,
i.e., if for each A ⊂ X we have nw(cl(A)) ≤ max{|A| , ω}.
A separable space of uncountable network weight (in particular, the Sor-
genfrey line) is an example of a space that is not ω-monolithic.
Example 3.1. The following spaces are monolithic:
(1) metrizable spaces;
(2) cosmic spaces;
(3) Σ-products of spaces with a countable base.
The union of two monolithic spaces need not be monolithic. The Nyemit-
skii plane may serve as an example. It is separable but has no countable
3. MONOLITHIC AND STABLE SPACES IN Cp-DUALITY 5
network, and hence is not ω-monolithic. At the same time it is the union
of two metrizable (hence monolithic) subspaces, one of which is closed and
discrete while the other is an open set of type Fσ.
Proposition 3.2. (1) The union of a locally finite family of closed
monolithic subspaces is monolithic;
(2) the property of (κ-) monolithicity is inherited by arbitrary subspaces;
(3) the product of a countable family of monolithic spaces is monolithic;
(4) every closed and continuous image of a monolithic space is mono-
lithic.
Given an infinite cardinal κ, a space X is called κ-stable if for every
continuous image Y of X the following conditions are equivalent:
(1) iw(Y ) ≤ κ;
(2) nw(Y ) ≤ κ.
It is well known that iw(Y ) ≤ nw(Y ) for every space Y . The converse
does not always hold, an example of this is the Sorgenfrey line. A discrete
space of cardinality c is not ω-stable: it can be condensed onto the space R,
which has a countable base.
A space X is called stable if it is κ-stable for every infinite cardinal κ. It
can be easily seen that X is stable if and only if for every continuous image
Y of X we have iw(Y ) = nw(Y ).
Theorem 3.3. (1) Every compact space is stable;
(2) every Lindelöf Σ-space is stable;
(3) any product of second countable spaces is stable;
(4) every Σ-product of second countable spaces is stable;
(5) each σ-product of second countable spaces is stable;
(6) any pseudocompact space is ω-stable;
(7) any Lindelöf P -space is ω-stable.
Proposition 3.4. (1) Every space X is κ-stable for κ ≥ nw(X);
(2) every continuous image of a stable space is stable;
(3) stability is inherited by open-closed subspaces;
(4) the union of a countable set of stable subspaces is stable.
There exists [32] a countably compact space X whose square contains an
open-closed uncountable discrete subspace Y . Then X is ω-stable, while Y is
not. Hence, (see Proposition 3.4, (3)) X ×X is not ω-stable, i.e., ω-stability
is not preserved, in general, under the transition of a square of a space.
A discrete space D of cardinality c is not ω-stable, but is homeomorphic
to a closed subspace of Rc. Thus, stability is not inherited, in general, by
closed subspaces (see Theorem 3.3, (3)).
Theorem 3.5. If the Hewitt realcompactification υX of a space X is an
ω-stable space, then X is ω-stable.
6 I. PRELIMINARIES
At first glance, monolithicity and stability appear to be rather unrelated
to each other. The following propositions indicate that there is a strong
relation between them:
Theorem 3.6. Cp(X) is κ-monolithic if and only if X is κ-stable.
Theorem 3.7. Cp(X) is κ-stable if and only if X is κ-monolithic.
Corollary 3.8. For any space X:
(1) X is monolithic if and only if Cp(X) is stable;
(2) X is stable if and only if Cp(X) is monolithic;
(3) Cp(Cp(X)) is monolithic if and only if X is monolithic;
(4) X is stable if and only if Cp(Cp(X)) is stable.
4. D-property
A neighborhood assignment for a space (X, τ) is a function φ : X → τ
with x ∈ φ(x) for every x ∈ X. X is said to be a D-space if for every
neighborhood assignment N , one can find a closed discrete subspace D of X
such that X =
⋃
{φ(x) : x ∈ D}.
The notion of a D-space seems to have had its origins in an exchange
of letters between E.K. van Douwen and E. Michael in the mid-1970s, but
the first paper on D-spaces is a 1979 paper by E.K. van Douwen and W.
Pfeffer [25]. The D-property is a kind of covering property; it is easily seen
that compact spaces and also σ-compact spaces are D-spaces, and that any
countably compact D-space is compact. Part of the fascination with D-
spaces is that, aside from these easy facts, very little else is known about
the relationship between the D-property and many of the standard covering
properties. For example, it is not known if a very strong covering property
such as hereditarily Lindelöf impliesD, and yet for all we know it could be that
a very weak covering property such as submetacompact or submetalindelöf
implies D. While these questions about covering properties remain unsettled,
there nevertheless has been quite a lot of interesting recent work on D-spaces.
The space [0, ω1) being countably compact and not compact is not a D-
space. It is an open question whether the union of two arbitrary D subspaces
is a D-space. Since every discrete space is a D-space, clearly, this property
is not preserved by continuous functions. Also, there exist [23] a Lindelöf
D-space B and a metrizable separable space M (hence a D-space) such that
B×M is not D, so D-property is not preserved, in general, by finite products.
Some facts about D-spaces are the following [12]:
Theorem 4.1. (1) Every closed subspace of a D-space is a D-space;
(2) the closed continuous image of a D-space is a D-space;
(3) the perfect inverse image of a D-space is a D-space.
Recall that the extent, e(X) of a space X is the supremum of the car-
dinalities of its closed discrete subsets, and the Lindelöf degree, l(X), is the
4. D-PROPERTY 7
least cardinal κ such that every open cover of X has a subcover of cardinality
κ. Note that e(X) ≤ l(X) for any X. It is easy to see that if X is a D-space
and U is an open cover with no subcover of cardinality < κ, then there must
be a closed discrete subset of X of size κ; hence e(X) = L(X). Since closed
subspaces of D-spaces are D, we have the following result:
Proposition 4.2. If X is a D-space and Y is a closed subspace of X,
then e(Y ) = l(Y ).
Corollary 4.3. Any countably compact D-space is compact.
A useful example to know, because it illustrates some of the limits of which
properties could imply D, is an old example due to E.K. van Douwen and
H. Wicke. There is a space Γ [24] which is non-Lindelöf and has countable
extent, and is Hausdorff, locally compact, locally countable, separable, first
countable, submetrizable, σ-discrete, and realcompact. This space is not D
because e(X) = ω < l(X).
For a time, all known examples of non-D-spaces failed to be D because
the conclusion of Proposition 4.2 failed. This was noted by W. Fleissner,
essentially bringing up the question: IsX aD-space if and only if e(Y ) = l(Y )
for every closed subspace Y ofX? R. Buzyakova [16] asked a related question:
Is X hereditarily D if and only if e(Y ) = l(Y ) for every Y ⊂ X? After some
consistent examples P. Nyikos [47] found a ZFC example of a non-D-space
in which e(Y ) = l(Y ) for every subspace Y , closed or not.
As we have mentioned, it is clear that compact spaces, in fact σ-compact
spaces, are D-spaces. Beyond this it often appears that some base or com-
pleteness “structure” is needed in order to prove certain spaces are D-spaces.
The following result shows how connections betweenD-spaces and generalized
metric spaces have been extensively studied.
Theorem 4.4. The following are D-spaces:
(1) Menger spaces [8];
(2) semistratifiable spaces (and hence Moore, semimetric, stratifiable,
and σ-spaces) [12] (see also [30]);
(3) subspaces of symmetrizable spaces [15];
(4) strong Σ-spaces (hence paracompact p-spaces) [18];
(5) protometrizable spaces (hence nonarchimedean spaces) [13];
(6) spaces having a point-countable base [6];
(7) spaces having a point-countable weak base [15] (see also [52]);
(8) sequential spaces with a point-countable w-system [15] or spaces with
a point-countable k-network [51];
(9) spaces with an ω-uniform base [8];
(10) base-base paracompact spaces (hence totally paracompact spaces) [59]
(see also [57]);
(11) t-metrizable spaces [39];
8 I. PRELIMINARIES
(12) spaces having a σ-cushioned (mod k) pair-network (hence Σ♯-spaces)
[46];
(13) spaces satisfying well-ordered (A), linearly semistratifiable spaces,
and elastic spaces [67].
Lin’s result (item (12)) simultaneously generalizes the statements “semis-
tratifiable implies D” and “strong Σ implies D”. The result of D. Soukuop
and X. Yuming (item (13)) about elastic spaces is an explanation of “pro-
tometrizable impliesD”, and their result about linearly semistratifiable spaces
generalizes “semistratifiable implies D”. Aurichi’s result about Menger (item
(1)) spurred much activity, in spite of the fact that it could be considered a
corollary of the previously known result that totally paracompact spaces are
D (item (10)).
Unfortunately, we still know very little about how the D-property affects
a space. Thus, more statements in the form “If X has the D-property then
...” would improve our understanding of this puzzling notion.
Finally, for completeness we present the definitions of the above concepts.
We say that a spaceX is aMenger space if for every sequence {Un : n ∈ ω}
of open covers there is {Vn : n ∈ ω} such that each Vn is a finite subset of Un
and X =
⋃
{
⋃
Vn : n ∈ ω}.
A space X is semistratifiable if to each open set U in X we can assign a
sequence {F (U, n) : n ∈ ω} of closed sets in X such that U =
⋃
{F (U, n) :
n ∈ ω} and F (U, n) ⊂ F (V, n) whenever U ⊂ V .
Suppose X is a topological space and d : X ×X → [0,∞) such that, for
all (x, y) ∈ X × X, d(x, y) = d(y, x) and d(x, y) = 0 if and only if x = y.
The function d is said to be a symmetric for X provided: for all nonempty
A ⊂ X, A is closed in X if and only if inf{d(x, z) : z ∈ A} > 0 for every
x ∈ X \A. In this case, one could say X is symmetrizable with symmetric d.
A space X is a strong Σ space if there exist a σ-locally finite family N of
closed sets in X and a cover K of X by compact subsets, such that N is a
network for X modulo K.
A base B for a space X is said to be an orthobase if whenever B′ is a subset
of B, either
⋂
B′ is open, or B′ is a local base for any point in
⋂
B′. A space
X is said to be proto-metrizable if it is paracompact and has an orthobase.
A weak base for a space X is a collection of subsets B = {Bx : x ∈ X}
where, for all x ∈ X, x ∈
⋂
Bx, Bx is closed under finite intersections and B
determines the topology on X in the following way: A set U ⊂ X is open in
X if and only if for all z ∈ U , there exists B ∈ Bz with B ⊂ U .
A collection N of subsets of X is a k-network if N is a network for X
modulo the family K(X) of all nonempty compact subsets of X.
Let X be a sequential space. If x ∈ W ⊂ X we say W is a weak neigh-
borhood of x if whenever {xn : n ∈ ω} converges to x then {xn : n ∈ ω} is
eventually in W , that is, |{xn : n ∈ ω} \W | < ω. A collection W of subsets
of X is said to be a w-system for the topology if whenever x ∈ U ⊂ X, with
4. D-PROPERTY 9
U open, there exists a subcollection V ⊂ W such that x ∈
⋂
V ,
⋃
V is a weak
neighborhood of x, and
⋃
V ⊂ U
We say that a base B for a topological space X is ω-uniform if for every
x ∈ X and every B ∈ B such that x ∈ B, the set {A ∈ B : x ∈ A and A 6⊂ B}
is countable.
A space X is base-base paracompact if X has an open base B such that
every base B′ ⊂ B contains a locally finite subcover.
A space (X, τ) is t-metrizable if there exists a metrizable topology π on
X with τ ⊂ π and an assignment H → JH from [X]<ω to [X]<ω such that
clτ (A) ⊂ clπ (
⋃
{JH : H ∈ [A]<ω}).
A collection P of pairs of subsets of a spaceX is called a pair-network forX
if whenever x ∈ U with U open in X, x ∈ P1 ⊂ P2 ⊂ U for some (P1, P2) ∈ P .
A collection P of pairs of subsets of a space X is called cushioned if for each
P ′ ⊂ P we have cl(
⋃
{P1 : (P1, P2) ∈ P ′}) ⊂ cl(
⋃
{P2 : (P1, P2) ∈ P ′}) .
A collection P of pairs of subsets of a space X is called a (modk)-network
for X if, there is a cover K of compact subsets of X such that, whenever
K ∈ K and K ⊂ U with U open in X, then K ⊂ P1 ⊂ P2 ⊂ U for some
(P1, P2) ∈ P .
Let X be a space. A collection P of ordered pairs P = (P1, P2) of subsets
of X is called a pair-base provided that P1 is open for all P ∈ P and that
for every x ∈ X and open set U containing x, there is a P ∈ P such that
x ∈ P1 ⊂ P2 ⊂ U .
Let X be a space, α an ordinal. We say that X satisfies (αA) if and
only if there is W = {W(x) : x ∈ X}, where W(x) = {W (β, x) : β < α},
such that x ∈ W (β, x) ⊂ X with the following property. For every open U
containing x, there exists an open set V (x, U) containing x and an ordinal
β = ϕ(x, U) < α such that x ∈ W (β, y) ⊂ U for all y ∈ V (x, U). If, in
addition, W (β, x) ⊂ W (γ, x) whenever γ < β < α, then we say that X
satisfies well-ordered (αA).
X is said to be semistratifiable over α (for some ordinal α) or linearly
semistratifiable if there exists a mapping F : α× τ(X) → CL(X) such that:
(1) U =
⋃
{F (U, β) : β < α} for all U ∈ τ(X);
(2) if U ⊂ W then F (U, β) ⊂ F (W,β) for all β < α;
(3) if γ < β < α, then F (U, γ) ⊂ F (U, β) for all U ∈ τ(X).
A space X is elastic if there is a pair-base P on X and transitive relation ≤
on P such that
(1) if P, P ′ ∈ P are such that P1 ∩ P
′
1 6= ∅ then P ≤ P ′ or P ′ ≤ P ;
(2) if P ∈ P and P ′ ⊂ {P ′ ∈ P : P ′ ≤ P} then cl(
⋃
{P ′
1 : P ′ ∈ P ′}) ⊂
⋃
{P ′
2 : P
′ ∈ P ′}.
10 I. PRELIMINARIES
5. D-property and metalindelöf spaces
In their article in Open Problems in Topology II, M. Hrušák and J.T.
Moore [41] list twenty central problems in set theoretic topology; the ques-
tion whether or not Lindelöf spaces are D is Problem 15 on this list, and
is attributed to E.K. van Douwen. T. Eisworth’s survey article [28], in the
same volume, lists several related questions. In fact, it is not known if any
of the following covering properties, even if you add “hereditarily”, imply
D: Lindelöf, paracompact, ultraparacompact, strongly paracompact, meta-
compact, metalindelöf, subparacompact, submetacompact, submetalindelöf,
paralindelöf, screenable, σ-metacompact (see [14] for definitions). Since the
non-D-space Γ, mentioned in the previous section, is σ-discrete, it follows
that weakly submetacompact does not imply D.
We know that every compact T1-space is a D-space. Recently, D.T.
Soukup and P.J. Szeptycki have shown that under ♦ there is a Hausdorff
hereditarily Lindelöf space which is not D [66]. However, it is not known
whether every regular Lindelöf space is D. In this section, we show that if X
is a metalindelöf locally Lindelöf D-space, then X is Lindelöf, this result was
proved independently by L.-Xue Peng and Hui Li [56].
The following notation will be used throughout: If φ is a neighborhood
assignment on X, and D ⊂ X, we let φ(D) =
⋃
{φ(x) : x ∈ D}.
A space X is metalindelöf if every open cover of X has point-countable
open refinement.
W. Fleissner and A. Stanley [30] introduced the notion of φ-sticky for a
neighborhood assignment φ, a tool which simplifies many D-space arguments.
G. Gruenhage [35] introduced the notions of nearly good relation and φ-sticky
modulo a relation as tools for proving that spaces have the D-property.
Let X be a space. We say that a relation R on X (respectively, from X to
[X]<ω) is nearly good if x ∈ cl(A) implies xR y for some y ∈ A (respectively,
xR ỹ for some ỹ ∈ [A]<ω).
Further, if φ is a neighborhood assignment on X, X ′ ⊂ X, and D ⊂ X, we
say D is φ-sticky mod R on X ′ if whenever x ∈ X ′ and xR y for some y ∈ D
(respectively, xR ỹ for some ỹ ∈ [D]<ω), then x ∈ φ(D). (In other words, it
means that φ(D) contains all the “relatives” of members (respectively, finite
subsets) of D that are in X.) We say more briefly that D is φ-sticky mod R
if D is N -sticky mod R on X.
The following proposition was essentially proved in [35].
Proposition 5.1. Let φ be a neighborhood assignment on X, and let R be
a nearly good relation on X. Suppose that given any countable closed discrete
D and nonempty closed F ⊂ X \ φ(D) such that D is φ-sticky mod R on F ,
there is a countable non-empty closed discrete E ⊂ F such that D ∪ E is φ-
sticky mod R on F . Then there is a closed discrete D∗ in X with φ(D∗) = X.
Lemma 5.2. If X has a point-countable open cover U such that U is
Lindelöf D for each U ∈ U then X is a D-space.
5. D-PROPERTY AND METALINDELÖF SPACES 11
Proof. Let φ : X → τ(X) be a neighborhood assignment on X. We
define a relation R on X as follows:
R = {(x, y) ∈ X ×X : {x, y} ⊂ U for some U ∈ U}.
Clearly, R is nearly good. We will show that conditions in Proposition 5.1
are satisfied. Let D be a countable closed discrete subspace of X and let
F ⊂ X \ φ(D) be a nonempty closed set such that D is φ-sticky mod R
on F . Take a family {Ωn : n ∈ ω} of infinite disjoint subsets of ω such
that ω =
⋃
{Ωn : n ∈ Ω} and {0, . . . , n} ⊂
⋃
{Ωk : k = 1, . . . , n}. We will
construct a set E in a recursive process.
Step 0. Let V ∗ ∈ U such that V ∗ ∩ F \ φ(D) 6= ∅. Let Y0 = cl(V ∗) ∩
F \ φ(D). Since D-property is inherited by closed subspaces, then Y0 is
Lindelöf D. Let E0 be a countable closed and discrete subspace of Y0 such
that Y0 ⊂ φ(E0). Since U is point-countable, then U0 = {U ∈ U : U∩E0 6= ∅}
is countable. Let {Um : m ∈ Ω0} be a numeration of U0.
Suppose that for each k ≤ n we have constructed countable closed sets
Ek ⊂ F in such form that
(1) the families Uk = {U ∈ G : U ∩Ek 6= ∅} are numbered by {Um : m ∈
Ωk};
(2)
⋃
{Ui ∩ F : i < k} ⊂ φ(
⋃
{D ∪ Ei : i ≤ k});
(3) Ek ⊂ F \ φ(
⋃
{D ∪ Ei : i < k}).
Step n + 1. If for each open set U in the family
⋃
{Uk : k ≤ n} we
have U ∩ F ⊂ φ(
⋃
{D ∪ Ek : k ≤ n}), take E =
⋃
{Ek : k ≤ n} and stop
the construction. It is not difficult to verify that in this case E satisfies the
required conditions. In the other case, let U∗ be the first set in
⋃
{Uk : k ≤ n}
which satisfies U∗ ∩ F \ φ(
⋃
{D ∪ Ek : k ≤ n}) 6= ∅. Let Yn+1 = cl(U∗) ∩ F \
φ(
⋃
{D ∪ Ek : k ≤ n}), then Yn+1 is Lindelöf D. Let En+1 be a countable
closed and discrete subset of Yn+1 such that Yn+1 ⊆ φ(En+1). Since U is
point-countable, then Un+1 = {U ∈ U : U ∩ En+1 6= ∅} is countable. Let
{Um : m ∈ Ωn+1} be a numeration of Un+1. Notice that by condition (2) in
step n, the election of U∗ and the fact that {0, . . . , n} ⊂
⋃
{Ωk : k = 1, . . . , n},
we have
⋃
{Ui : i < n+ 1} ⊂ φ(
⋃
{Ei : i ≤ n+ 1}). Therefore (1),(2) and (3)
hold in step n+ 1.
In order to finish the construction let E = ∪{En : n ∈ ω}. We will
show that E satisfies the conditions in Proposition 5.1. Clearly, E ⊂ F . Let
W =
⋃
{
⋃
Un : n ∈ ω} =
⋃
{Uk : k ∈ ω}. For each k ∈ ω, by condition (2) in
step k + 1, Uk ∩ F ⊂ φ(E). Therefore W ∩ F ⊂ φ(E). Suppose that x ∈ F
and xR y for some y ∈ D ∪E. If y ∈ D, since D is φ-sticky mod R on F , we
have x ∈ φ(D) ⊂ φ(D ∪E). If y ∈ E then {x, y} ∈ U ∩ F = Uk ∩ F for some
k ∈ ω. Therefore, y ∈ Uk ∩ F ⊂ φ(E) ⊂ φ(D ∪E). This shows that D ∪E is
φ-sticky mod R on F . We shall prove that E is a closed and discrete subset
of X.
Let x ∈ cl(E) ⊂ F . Since R is nearly good, then xR y for some y ∈ E.
Since D ∪ E is φ-sticky mod R on F , then x ∈ φ(D ∪ E) ∩ F . Notice that
12 I. PRELIMINARIES
x 6∈ φ(D). Let n be the first natural number such that x ∈ φ(
⋃
{D∪Ek : k ≤
n}), then by condition (3), x ∈ En. Since E1 ∪ ... ∪ En is closed and discrete
we can find an open neighborhood Vx of x with Vx ∩ (E1 ∪ . . . ∪ En) = {x},
then by construction, Ux = Vx ∩ φ(E1 ∪ ... ∪ En) is an open neighborhood of
x with Ux ∩ E = {x}. This shows that E is closed and discrete.
Finally, by Proposition 5.1, there exists a closed and discrete subspace D∗
of X such that N(D∗) = X. Therefore, X is a D-space. 
A space X is called locally Lindelöf D if every point x of X has a neigh-
borhood Ux which is a Lindelöf D-space.
Theorem 5.3. If X is a metalindelöf locally Lindelöf D-space, then X
has the D-property.
Proof. For each x ∈ X let Ux be an open neighborhood of x such that
Ux is Lindelöf D. If U is a point-countable open refinement of {Ux : x ∈ X}
then U satisfies the hypothesis of proposition 5.2. Hence X is D. 
Some spaces which have certain covering properties and scattered property
are D-spaces.
Let K denote a class of spaces which are hereditary with respect to closed
subspaces. A space X is called a K-scattered space if for any nonempty closed
subset F of X, there is some x ∈ F such that the point x has a neighborhood
Ux in the subspace F such that Ux ∈ K.
The class of all D-spaces is denoted by D and the class of all compact
spaces is denoted by C. Clearly, every D-scattered space is C-scattered space
and any scattered space is D-scattered. Every submetacompact D-scattered
space is a D-space [54]. Thus, every submetacompact C-scattered space is
D. However, it is not known whether every regular metalindelöf scattered
space is a D-space.
In [53] L.-Xue Peng, independently, gave a proof of Lemma 5.2 and used
this to prove the following:
Theorem 5.4. Let X be a metalindelöf space.
(1) If X is C-scattered space with a finite rank, then X is a D-space.
(2) If X is D-scattered space with locally countable extent, then X is a
D-space.
It is proved in [9] that every T1 Lindelöf space which is the union of
less than cov(M) compact subsets is a D-space, where M is the ideal of
meager subsets of the real line. Since MA+¬CH implies ω1 < c = cov(M),
it is consistent that every T1 Lindelöf space of cardinality ω1 is a D-space.
G. Gruenhage [36] asked if it is consistent that every paracompact space of
cardinality ω1 is a D-space. Recently Hang Zhang and Wei-Xue Shi have
proved that:
Theorem 5.5 (MA+¬CH). [79] Every regular T1 submetalindelöf space
of cardinality ω1 is D.
6. D-PROPERTY IN FUNCTION SPACES 13
6. D-property in function spaces
Recall that for a compact space X, Baturov’s theorem [10] states that
l(Y ) = e(Y ) for every subspace Y of Cp(X). M. Matveev asked whether the
conclusion in the Baturov theorem for compacta can be strengthened to the
D-property. R. Buzyakova answered this question; indeed; she proved the
following result [16]:
Theorem 6.1. Let X be compact and Y ⊂ Cp(X). Then Y is a D-space.
Later, another proof was obtained by H. Guo and H.J.K. Junnila. They
proved that t-metrizable spaces are hereditarily D-spaces (see Theorem 4.4
(11)). Since Cp(X) is t-metrizable for compact K (indeed, the sup-norm
topology π and the pointwise topology τ of C(X) witness the t-metrizability
of Cp(X), see [26]), Buzyakova’s result is a consequence of Guo and Junnila’s
result.
As we have seen, l(X) = e(X) for every D-space X and every count-
ably compact D-space is compact. This simple observation leads us to the
following corollaries of Theorem 6.1, which are, in fact, well known classical
theorems.
Corollary 6.2 (Baturov’s Theorem for compacta). Let X be compact.
Then l(Y ) = e(Y ) for every subspace Y of Cp(X).
Corollary 6.3 (Grothendieck’s Theorem for Compacta [34]). Let X be
compact and Y a countably compact subspace of Cp(X). Then Y is compact.
Since Baturov’s theorem holds not only for compacta but for all Lindelöf
Σ-spaces, R. Buzyakova supposed that the following question might have a
chance for an affirmative answer (This question was also suggested by M.
Matveev, see [35]): Let X be a Lindelöf Σ space, it is true that Cp(X) is a
hereditary D-space?
In [35] G. Gruenhage introduced the concept of nearly good relation and
φ-sticky modulo a relation (see §5). He proved the next result, using elemen-
tary submodels.
Given a neighborhood assignment φ on X, let us call a subset Z of X
φ-close if for every x, y ∈ Z we have x ∈ φ(y) (equivalently, Z ⊂ φ(x) for
every x ∈ Z).
Proposition 6.4. Let φ be a neighborhood assignment on X. Suppose
there is a nearly good R on X (respectively, from X to [X]<ω) such that for
any y ∈ X (respectively, ỹ ∈ [X]<ω), R−1(y) \ φ(y) (respectively, R−1(ỹ) \
⋃
φ(ỹ)) is the countable union of φ-close sets. Then there is a closed discrete
D such that φ(D) = X.
Remark 6.5. Note that if φ and R satisfy the hypotheses of the proposi-
tion, then so does their restriction to any subspace. So, if for any φ on X we
can produce such an R, then X is hereditarily D.
14 I. PRELIMINARIES
Using Proposition 6.4, G. Gruenhage [35] gave a short proof of Theorem
4.4 (6) and Theorem 6.1. Also, using Proposition 6.4, answered the above
question posed by M. Matveev by proving that:
Proposition 6.6. Let X be a Lindelöf Σ space. Then Cp(X) is hereditarily
D.
In another direction, R. Buzyakova proposed the next question [16]: Let
X be a countably compact space, is it true that every subspace of Cp(X)
is a D-space? In [19], she described a counterexample. She considered the
space X = {α ≤ ω2 : cf(α) 6= ω1} and proved that l(Cp(X)) = ω2 while
e(Cp(X)) = ω. This example also answers Reznichenko’s question (whether
Baturov’s theorem holds for countably compact spaces) in the negative.
R. Buzyakova also proved that:
Theorem 6.7. [19] Cp(X) is Lindelöf for any first countable countably
compact subspace of an ordinal.
Since the question whether or not Lindelöf spaces are D remains open
even in the class of Cp-spaces, in search of a counterexample (if there exists
one) R. Buzyakova proposed the following question: Is Cp(X) a D-space for
any first countable countably compact subspace of an ordinal? L.-Xue Peng
answered this question by proving the following result.
Theorem 6.8. [52] Suppose that X is a first countable countably compact
subspace of an ordinal. Then Cp(X) is a D-space.
V.V. Tkachuk also solved the above question with an interesting approach.
Recall that a space X is Sokolov if for any sequence {Fn : n ∈ ω} where
Fn is a closed subset of Xn for every n ∈ N , there exists a continuous map
f : X → X such that nw(f(X)) = ω and fn(Fn) ⊂ Fn for all n ∈ ω.
V.V. Tkachuk proved the following result and showed that his technique
provides a different method to prove Theorem 6.8.
Theorem 6.9. [74] Suppose that X is a countably compact first countable
subspace of an ordinal. Then X is a Sokolov space.
It is known that [72]: (1) For any Sokolov first countable space X, the
space Cp(X,E) is Lindelöf whenever E is a second countable space. (2) The
countable power of a Sokolov space is a Sokolov space. (3) If X is Sokolov,
then Cp,n(X) is also Sokolov for any n ∈ ω. (4) For any Sokolov first countable
space X, the space Cp,n(X) is Lindelöf for any n ∈ ω. Thus, we obtain the
following consequences of Theorem 6.9 which are explanations of Theorem
6.7.
Corollary 6.10. [74] If X is a first countable countably compact subspace
of an ordinal, then (Cp(X))ω and Cp(X
ω) are Lindelöf spaces.
Corollary 6.11. [74] If X is a first countable countably compact subspace
of an ordinal, then the iterated function space Cp,2n+1(X) is Lindelöf for any
n ∈ ω.
7. MONOTONE MONOLITHICITY 15
7. Monotone monolithicity
Monotone (covering) properties have been intensively studied in several
contexts. Monotone normality, due to R.W. Heath, D.J. Lutzer, and P. Zenor,
is a classical property in the study of generalized metric spaces. Stratifiable
spaces, introduced by J. Ceder [20], are precisely the class of monotonically
perfectly normal spaces. Monotonically compact and monotonically Lindelöf
spaces were defined by M. Matveev and were first studied in print by H.
Bennett, D. Lutzer and M. Matveev in [11]. P.M. Gartside and P.J. Moody
[31] defined monotonically paracompact spaces and showed that they are ex-
actly the class of protometrizable spaces. S.G. Popvassilev [58] introduced
the notion of monotonically (countably) metacompact space. Monotonic ver-
sions of countable paracompactness and countable metacompactness, that are
quite different in spirit from the monotonic properties before mentioned, were
introduced independently in [33], [50], and [69].
There is an interesting relation between these properties andD-spaces. As
we saw in Theorem 4.4, monotonically perfectly normal (stratifiable) spaces
and monotonically paracompact (protometrizable) spaces are D-spaces. Re-
cently, in [56], it was proved that every monotonically (countably) meta-
compact space is hereditarily a D-space. On the other hand, it is an open
question if every paracompact monotonically normal space is a D-space [36].
This section is devoted to monotone monolithicity, a relatively new mono-
tone property. Of course, we have that any monotonically monolithic space
is hereditarily a D-space.
V.V. Tkachuk, introduced the concept of monotonically monolithic space.
Definition 7.1. [77] Given a set A in a space X we say that a family N
of subsets of X is an external network of A in X if for any x ∈ A and an open
set U ⊂ X with x ∈ U there exists N ∈ N such that x ∈ N ⊂ U .
Definition 7.2. [77] Say that a space X is monotonically monolithic if,
for any A ⊂ X we can assign an external network O(A) to the set cl(A) in
such a way that the following conditions are satisfied:
(1) |O(A)| ≤ max{|A| , ω};
(2) if A ⊂ B ⊂ X then O(A) ⊂ O(B);
(3) if α is an ordinal and we have a family {Aβ : β < α} of subsets of
X such that β < β′ < α implies Aβ ⊂ Aβ′ then O(
⋃
Aβ : β < α) =
⋃
{O(Aβ) : β < α}.
The class of monotonically monolithic spaces seems to be interesting in
itself. This class has nice categorical properties and contains some important
classes of topological spaces.
Theorem 7.3. Monotone monolithicity is preserved by:
(1) countable products [77];
(2) σ-products [1];
16 I. PRELIMINARIES
(3) arbitrary subspaces [77];
(4) closed maps [77].
Proposition 7.4. The following spaces are monotonically monolithic:
(1) spaces with a unique non-isolated point [77];
(2) spaces with a point-countable base [77];
(3) t-metrizable spaces [39];
(4) stratifiable spaces [38].
Among the main results obtained in [77] we found the following.
Proposition 7.5. Cp(X) is monotonically monolithic for X Lindelöf Σ.
Theorem 7.6. Any monotonically monolithic space is hereditarily D.
Notice that Proposition 7.5 is a generalization of Proposition 6.6. From
Proposition 7.4 we have, in particular, every metrizable space is monoton-
ically monolithic. Therefore, the class of monotonically monolithic spaces
is relatively large. Also, it is easy to see from the definition that cosmic
spaces are monotonically monolithic. Notice that every monotonically mono-
lithic space is monolithic. However, not every compact monolithic space is
monotonically monolithic; indeed, the space [0, ω1] is compact and monolithic
but not monotonically monolithic; in fact, its subspace [0, ω1) is not D (see
Theorem 7.6). Uncountable products of monotonically monolithic spaces not
need be monotonically monolithic; Rω1 is not monotonically monolithic since
[0, ω1) embeds as a subspace of it.
We now give a characterization of monotonically monolithic spaces which
has served to prove several interesting results.
Theorem 7.7. [38] A space X is monotonically monolithic if and only if
one can assign to each finite subset F of X a countable collection N (F ) of
subsets of X such that, for each A ⊂ X, the family
⋃
{N (F ) : F ∈ [A]<ω}
contains a network for cl(A).
It is well known and easy to prove that any compact space of countable
tightness is a Fréchet-Urysohn space. The following results show that in the
presence of monotonic monolithicity this result can be strengthened.
Theorem 7.8. [77] If a countably compact space X is monotonically
monolithic then X is a Fréchet-Urysohn compact space.
It follows from the previous result and Corollary to 3.25 in [42] that any
compact monotonically monolithic space is separable, therefore:
Corollary 7.9. [77] If a compact space X is monotonically monolithic
and ω1 is a caliber of X then X is metrizable.
There exists a Lindelöf monotonically monolithic space of uncountable
tightness. Indeed, let Lω1
be the one-point Lindelöfication of a discrete space
of cardinality ω1, then Lω1
is monotonically monolithic by Proposition 7.4 (1)
and it is easy to see that it is a Lindelöf space of uncountable tightness.
8. WEAKLY MONOTONE MONOLITHIC SPACES 17
8. Weakly monotone monolithic spaces
L.-Xue Peng generalized the concept of monotonically monolithic space.
He introduced the concept of weakly monotonically monolithic space and
show that every such space has the D-property.
Definition 8.1. [55] We say that a spaceX is weakly monotonically mono-
lithic if for any A ⊂ X we can assign an external network O(A) of A in such
a way that the following conditions are satisfied:
(1) |O(A)| ≤ max{|A| , ω};
(2) if A ⊂ B ⊂ X then O(A) ⊂ O(B);
(3) if α is an ordinal and we have a family {Aβ : β < α} of subsets of
X such that β < β′ < α implies Aβ ⊂ Aβ′ then O(
⋃
Aβ : β < α) =
⋃
{O(Aβ) : β < α};
(4) If A ⊂ X is not closed in X then there is some x ∈ cl(A) \ A such
that O(A) is an external network of {x}.
Theorem 8.2. [55] If X is a weakly monotonically monolithic space, then
X is a D-space.
G. Gruenhage observed that weakly monotone monolithicity has a char-
acterization similar to Theorem 7.7.
Theorem 8.3. [38] A space X is weakly monotonically monolithic if and
only if one can assign to each finite subset F of X a countable collection
N (F ) of subsets of X such that, for each non-closed set A ⊂ X, there is
some x ∈ cl(A) \ A such that
⋃
{N (F ) : F ∈ [A]<ω} is a network for {x}.
Let us recall that a family N of subsets of X is a cs∗-network of X, if for
any sequence {xn : n ∈ ω} which converges to a point x and for any open
set U which contains x, there is some N ∈ N such that x ∈ N ⊂ U and
|{n ∈ ω : xn ∈ N}| = ω.
Notice that every k-network of X (see §4) is a cs∗-network of X.
Suppose that N is a point-countable cs∗-network of X. For any A ⊂ X,
we let O(A) = {N ∈ N : N ∩ A 6= ∅}. If X is sequential, we see that O(A)
satisfies the conditions which appear in Definition 8.1. Thus:
Proposition 8.4. [55] If X is a sequential space with a point-countable
cs∗-network, then X is a weakly monotonically monolithic space.
Corollary 8.5. [55] If X is a sequential space with a point-countable cs∗-
network, then X is a D-space.
Since every k-network of X is a cs∗-network, the second part of Theorem
4.4 (8) is a consequence of Corollary 8.5.
In [38] G. Gruenhage used Theorem 7.7, Proposition 6.4 and Remark 6.5
to obtain a quick proof that monotonically monolithic spaces are hereditarily
D. We will describe his proof because it is short and elegant: Let X be
18 I. PRELIMINARIES
monotonically monolithic witnessed by operator N as in Theorem 7.7, and
let φ be a neighborhood assignment on X. Define a relation R from X to
[X]<ω, as
xRF if and only if x ∈ N ⊂ φ(x) for some N ∈ N (F ).
Since
⋃
{N (F ) : F ∈ [A]<ω} contains a network for cl(A), it is straightforward
to check that this R is nearly good. Now fix F ∈ [X]<ω. Let N(x) denote any
N ∈ N (F ) such that x ∈ N ⊂ φ(x) (if such N exists). Then X(N) = {x ∈
X : xRF and N(x) = N} is φ-close, and R−1(F ) =
⋃
{X(N) : N ∈ N (F )}.
So R−1(F ) is a countable union of φ-close sets, hence, X and R satisfy the
hypothesis in Proposition 6.4 and hence X is hereditarily D (see also Remark
6.5).
Let us call a relation R on X (resp., from X to [X]<ω) nearly OK if A
non-closed implies xR y for some x ∈ cl(A) \A and some y ∈ A (resp., xR ỹ
for some x ∈ cl(A) \ A and some ỹ ∈ [A]<ω).
G. Gruenhage [38] introduced the previous concept and described a slight
modification of the above argument to give a quick proof of Theorem 8.2.
9. Monotone κ-monolithicity
In [1] O. Alas, V.V. Tkachuk and R.G. Wilson continued the study un-
dertaken in [77]. They introduced the notion of monotone κ-monolithicity
for any infinite cardinal κ and showed that some theorems that were proved
for monotonically monolithic spaces also hold for monotonically κ-monolithic
spaces.
Definition 9.1. Given an infinite cardinal κ we say that a space X is
monotonically κ-monolithic if, for any A ⊂ X with |A| ≤ κ we can assign
an external network O(A) to the set cl(A) in such a way that the following
conditions are satisfied:
(1) |O(A)| ≤ max{|A| , ω};
(2) if A ⊂ B ⊂ X then O(A) ⊂ O(B);
(3) if λ < κ is a cardinal and we have a family {Aα : α < λ} ⊂ [X]≤κ
such that α < β < λ implies Aα ⊂ Aβ then O(
⋃
Aα : α < λ) =
⋃
{O(Aα) : α < λ}.
The following proposition shows some categorical behavior of monotone
κ-monolithicity.
Theorem 9.2. [1] Monotone κ-monolithicity is preserved by:
(1) countable products;
(2) σ-products;
(3) arbitrary subspaces;
(4) closed maps.
In [1] an example of a linearly orderable space which is monotonically ω-
monolithic but not monotonically ω1-monolithic was presented (the subspace
9. MONOTONE κ-MONOLITHICITY 19
of [0, ω2) of all ordinals of uncountable cofinality). On the other hand, it
is easy to see that any monotonically monolithic space is monotonically κ-
monolithic for any infinite cardinal κ. We can see that the converse is true.
Proposition 9.3. [1] A space is monotonically monolithic if and only if
it is monotonically κ-monolithic for any infinite cardinal κ.
In particular, all the spaces which appear in Proposition 7.4 are monotone
κ-monolithic for any cardinal κ.
As well as weakly monotonic monolithicity, monotonic κ-monolithicity has
a characterization similar to Theorem 7.7.
Theorem 9.4. [56], [62], [75] A space X is monotonically κ-monolithic if
and only if one can assign to each finite subset F of X a countable collection
N (F ) of subsets of X such that, for each A ⊂ X with |A| ≤ κ,
⋃
{N (F ) :
F ∈ [A]<ω} contains a network for cl(A).
In some cases κ-monolithicity implies monolithicity. It is a well known fact
that an ω-monolithic space of countable tightness is monolithic. In [56], [62]
and [75] it was proved independently (using Theorem 9.4) that an analogous
result holds for monotonic κ-monolithicity.
Proposition 9.5. If X is a monotonically κ-monolithic space such that
t(X) ≤ κ, then X is monotonically monolithic.
Let X be a space such that its Hewitt realcompactification υX is ω-stable.
By Theorem 3.5 the space X is itself ω-stable and by Theorem 3.6 the space
Cp(X) is ω-monolithic. The following result shows that a similar result holds
for monotone κ-monolithicity (see Theorem 3.3 (2)).
Theorem 9.6. [1] If X is a space such that its Hewitt κ-extension is
Lindelöf Σ, then Cp(X) is monotonically κ-monolithic.
It is well known [29] and easy to proof that if X is a pseudocompact
space then the Hewitt realcompactification υX of X and the Stone-Čech
compactification βX of X coincide. Thus, υX = βX is Lindelöf Σ for a
pseudocompact space X. Also, it is known that υX is Lindelöf Σ whenever
υ(Cp(X)) is Lindelöf Σ (see Theorem IV 9.5 in [3]). Thus, we have:
Corollary 9.7. [1] If X is pseudocompact then Cp(X) is monotonically
ω-monolithic.
Corollary 9.8. [1] If υCp(X) is Lindelöf Σ then Cp(X) is monotonically
ω-monolithic.
Since ω-monolithic spaces are more widely used that monolithic ones, it
is natural in the study of monotone monolithicity that for the results on
ω-monolithicity, there are some strong analogies for its stronger version.
20 I. PRELIMINARIES
Theorem 9.9. [1] Suppose that a monotonically ω-monolithic space X
is the union of at most ω1-many cosmic spaces. Then X is monotonically
monolithic.
Proposition 9.10. [75] If X is a monotonically ω-monolithic perfectly
normal compact space, then X is metrizable.
Proposition 9.11. [1] If a countably compact space X is monotonically
ω-monolithic, then X is compact and has the Fréchet-Urysohn property.
Notice that Proposition 9.11 is an explanation of Theorem 7.8. In [1] an
example of a monotonically ω1-monolithic pseudocompact space which is not
compact was presented. Also, an example of a compact scattered monolithic
space such that Cp(X) is Lindelöf and X is hereditarily D-space, but such
that X is not monotonically ω-monolithic, was presented.
In general, the fact that a space X has a weaker monotonically mono-
lithic topology does not imply that X must be κ-monolithic. To see that,
observe that the Sorgenfrey line is not ω-monolithic while it condenses onto
a monotonically monolithic space R. Therefore, even the hereditary Lin-
delöf property does not help to “lift” monolithicity under a condensation.
The situation is different if we look at Lindelöf Σ-spaces. Indeed, suppose
that f : X → Y is a condensation and Y is κ-monolithic. If A ⊂ X and
|A| ≤ κ, then cl(A) is a Lindelöf Σ space which condenses into the space
Z = cl(f(A)) with nw(Z) ≤ κ, so we can apply stability of cl(A) to conclude
that nw(cl(A)) ≤ κ, i.e., X is κ-monolithic. Therefore:
Proposition 9.12. [75] If a Lindelöf Σ-space X condenses onto a mono-
lithic space, then X is monolithic.
The same results hold if we replace monolithicity by monotone mono-
lithicity.
Proposition 9.13. [75] If a Lindelöf Σ-space X condenses onto a mono-
tonically monolithic space, then X is monotonically monolithic.
10. The Collins-Roscoe property
In [22], Collins and Roscoe investigated some conditions for metrizability.
In that article they introduced the following condition called condition (G)
that has been intensively studied in different contexts:
(G) To each x ∈ X, is assigned a countable collection G(x) of subsets of
X such that, whenever x ∈ U , where U is an open subset of X, there is an
open V with x ∈ V ⊂ U such that, whenever y ∈ V , then x ∈ N ⊂ U for
some N ∈ G(y).
If in addition all elements in G(y) are open, we say that X satisfies open
(G). G. Gruenhage observed that condition (G) is equivalent to the following
concept that recently has been called the Collins-Roscoe property.
10. THE COLLINS-ROSCOE PROPERTY 21
Definition 10.1. A space X has the Collins-Roscoe property if for each
x ∈ X, one can assign a countable collection G(x) of subsets of X such that,
for any A ⊂ X,
⋃
{G(x) : x ∈ A} contains an external network for cl(A).
Since the Collins-Roscoe property was introduced in the study of metriz-
ability, it is interesting to know the relation between this property and cov-
ering properties. In [45] the following result was proved.
Proposition 10.2. If the space X has the Collins-Roscoe property, then
X is hereditarily metalindelöf.
It follows immediately from the definition and Theorem 7.7 that every
space with the Collins-Roscoe property is monotonically monolithic. G. Gru-
enhage (see [35]) asked if the Collins-Roscoe property is equivalent to mono-
tonic monolithicity, and suggested that Cp(X) for some Lindelöf Σ-space X
might be a place to look for an example distinguishing the two concepts (see
Proposition 7.5).
V.V. Tkachuk [78] has shown that Cp(βD) does not have the Collins-
Roscoe property whenever D is an uncountable discrete space (but it is mono-
tonically monolithic). Indeed, if Dω1
is a discrete space of cardinality ω1, then
it is easy to see that Dω1
is a continuous image of D and hence Cp(Dω1
) is
homeomorphic to a closed subspace of Cp(D) (see Proposition 2.4). A. Dow,
H. Junnila, J. Pelant [27] proved that Cp(Dω1
) is not metalindelöf; as an im-
mediate consequence, the space Cp(D) is not metalindelöf either. However,
the Collins-Roscoe property in a space implies that it is metalindelöf (see
Proposition 10.2), so the space Cp(D) does not have this property.
In [78] Collins-Roscoe spaces were studied systematically; let us formulate
some categorical properties of the class of Collins-Roscoe spaces.
Theorem 10.3. Collins-Roscoe property is preserved by:
(1) arbitrary subspaces [22];
(2) countable products [78];
(3) closed maps [78];
(4) σ-products [78].
Since the Collins-Roscoe property is stronger than monotone monolithic-
ity, it is natural to expect that some questions which are open for monotone
monolithicity could be solved positively for the spaces with the Collins-Roscoe
property. Also, in the presence of the Collins-Roscoe property several results
which are true for monotonically monolithic spaces can be strengthened. Now
we present some results of this nature.
V.V. Tkachuk gave a generalization of Theorem 9.9 by showing that:
Theorem 10.4. [78] Suppose that a monotonically ω-monolithic space X
is the union of at most ω1-many cosmic spaces. Then X has the Collins-
Roscoe property and, in particular, X is hereditarily metalindelöf.
22 I. PRELIMINARIES
Corollary 10.5. [78] Assume that X is a monotonically ω-monolithic
space, d(X) ≤ ω1 and t(X) ≤ ω. Then X has the Collins-Roscoe property
and, in particular, X is hereditarily metalindelöf.
It is still an open question [1] whether any monotonically monolithic space
X has to be cosmic when ω1 is a caliber of X. It turns out that a positive
answer can be given for the spaces with the Collins-Roscoe property (see also
Corollary 7.9).
Theorem 10.6. [78] If a space X has the Collins-Roscoe property and ω1
is a caliber of X, then X is cosmic.
It turns out that, as monotonic monolithicity, the Collins-Roscoe property
is also “lifted” by condensations of Lindelöf Σ-spaces (see Proposition 9.13).
Proposition 10.7. [75] Suppose that a Lindelöf Σ-space X condenses
onto a Collins-Roscoe space. Then X is a Collins-Roscoe space.
11. Monotone monolithicity in compact spaces
Recall that a compact space X is said to be Eberlein compact if it em-
beds in Cp(Y ) for some compact space Y . A compact space X is Gul’ko
compact if Cp(X) is a Lindelöf Σ-space. A compact space X is said to be
Corson compact if X embeds in a Σ-product of real lines, i.e., in {x ∈ R
κ :
|{α ∈ κ : x(α) 6= 0}| ≤ ω} for some cardinal κ.
It is well known that any Eberlein compact space is Gul’ko compact and
that any Gul’ko compact space is Corson compact.
A corollary of Buzyakova’s result (Cp(X) is hereditarily a D-space when-
ever X is compact) is that Eberlein compact spaces are hereditarily D. This
prompted the natural question, due to Arhangel’skii, whether Corson com-
pact spaces are hereditarily D. G. Gruenhage showed that the answer is
positive.
Theorem 11.1. [36] Every Corson compact space is hereditarily a D-
space.
However, for Gul’ko compact spaces (hence for Eberlein compact spaces)
we can explain this result. Indeed, by Proposition 7.5 we have that every
Gul’ko compact space is monotonically monolithic. On the other hand, it
is well known that Corson compacts are monolithic (see Example 3.1 (3)).
The above suggests natural questions about the relationship of monotonically
monolithic spaces and Corson compacta.
O. Alas, V.V. Tkachuk and R.G. Wilson proposed the following questions
[1]:
(1) Is every Corson compact space monotonically ω-monolithic?
(2) Assume that X is a monotonically monolithic compact space, must
X be Corson compact?
11. MONOTONE MONOLITHICITY IN COMPACT SPACES 23
(3) Must every monotonically ω-monolithic compact space be mono-
lithic?
(4) Is it true that any monotonically monolithic linearly ordered compact
space is metrizable?
With respect to the first question, G. Gruenhage [35] gave a Corson com-
pact space which is not monotonically ω-monolithic, so the answer to the
first question is negative. Gruenhage’s construction is very difficult, but if
CH is assumed, then it is easy to obtain such an example. Indeed, in [75]
V.V. Tkachuk gave another method for constructing a Corson compact space
which is not monotonically ω-monolithic.
In [37], G.Gruenhague considered the following game G(H,X) of length
ω played in a space X, where H is a closed subset of X. There are two
players, O and P . In the nth round, O chooses an open superset On of H,
and P chooses a point pn ∈ On. We say O wins the game if pn → H in
the sense that every open superset of H contains pn for all but finitely many
n ∈ ω. G. Gruenhage showed that a compact Hausdorff space X is Corson
compact if and only if O has a winning strategy in G(∆, X2), where ∆ is the
diagonal in X2. Using this game characterization of Corson compact spaces
G.Gruenhaghe proved the following result which implies that the answer to
the second question above by O. Alas, V.V. Tkachuk and R.G. Wilson is
positive.
Proposition 11.2. [35] If X is monotonically ω-monolithic and countably
compact, then O has a winning strategy in G(H,X) for any closed subset H
of X.
Corollary 11.3. [35] If X is compact and monotonically ω-monolithic,
then X is a Corson compact space.
In [39] it was proved, using thick covers, that any monotonically mono-
lithic compact space is a Corson compact space.
As a consequence of Corollary 11.3, we can obtain an explanation of
Proposition 9.11. Let X be a monotonically ω-monolithic countably compact
space. By the previous result, X is compact and has the Fréchet-Urysohn
property. By Corollary 11.3, X is a Corson compact space. Since every
Corson compact has countable tightness, by Proposition 9.5, X is monotoni-
cally monolithic. Thus, we have the following result which answers the third
question above by O. Alas, V.V. Tkachuk and R.G. Wilson .
Proposition 11.4. [56], [62], [75] If X is a monotonically ω-monolithic
countably compact space, then X is a monotonically monolithic Corson com-
pact space and has the Fréchet-Urysohn property.
It is well known that any linearly ordered Corson compact space is metriz-
able, so the previous result also gives a positive answer to the fourth question
above by O. Alas, V.V. Tkachuk and R.G. Wilson .
24 I. PRELIMINARIES
K. Alster [2] proved that a scattered compact space is Corson compact if
and only if it is Eberlein compact. It was asked in [77] whether every scattered
monotonically monolithic compact space is Eberlein compact. It follows from
Corollary 11.3 and Alster’s result that any scattered monotonically monolithic
compact space is Eberlein compact.
Now we will present some results for the class of Gul’ko compact spaces
and some related conclusions.
Recall that a family U of subsets of a space X is T0-separating if for
any distinct points x, y ∈ X, there exists U ∈ U such that U ∩ {x, y} is a
singleton. A family U =
⋃
{Un : n ∈ ω} of subsets of X is called weakly
σ-point-finite if for any point x ∈ X we have the equality U =
⋃
{Un :
the family Un is point-finite at x}.
The following result was proved in [65].
Theorem 11.5. A compact space X is Gul’ko compact if and only if X
has a weakly σ-point finite T0-separating cover U =
⋃
{Un : n ∈ ω} by open
Fσ-sets.
G. Gruenhage used this characterization of Gul’ko compact spaces to
prove the following result which improve our understanding of Gul’ko com-
pact spaces.
Theorem 11.6. [35] If X is Gul’ko compact, then X has the Collins-
Roscoe property.
V.V. Tkachuk provides an alternative way to prove Theorem 11.6.
In [26], A. Dow, H. Junnila, and J. Pelant considered several properties
implied by the existence of a stronger metric topology on function spaces.
Among other things, they introduced spaces with point-countably expandable
networks. Recall that given a space X and a family A of subsets of X, we
say that a family of open sets E = {OA : A ∈ A} is an open expansion of A if
A ⊂ OA for any A ∈ A. A family A of subsets of a space X is point-countably
expandable if it has a point-countable open expansion in X.
Suppose that N is a network in X and {ON : N ∈ N} is a point-countable
open expansion of N . Given any point x ∈ X, let G(x) = {N ∈ N : x ∈ ON};
it is clear that the collection G(x) is countable. Take any set A ⊂ X and a
point x ∈ cl(A); for any open set U which contains x, there exists N ∈ N
such that x ∈ N ⊂ U . Pick a point a ∈ A ∩ ON , then N ∈ G(a). Thus, the
family
⋃
{G(a) : a ∈ A} contains an external network at all points of A, and
therefore:
Proposition 11.7. [75] If a space X has a point-countably expandable
network, then X is a Collins-Roscoe space.
The class of spaces with a point-countably expandable network is impor-
tant because it contains all Gul’ko compact spaces. Hence, Theorem 11.6 is
a corollary of Proposition 11.7.
11. MONOTONE MONOLITHICITY IN COMPACT SPACES 25
In view of Theorem 11.6, G. Gruenhage asked about the relationship
of monotonically monolithic (Collins-Roscoe) spaces and Gul’ko compacta.
Indeed, he proposed the following question: If X is compact, and has the
Collins-Roscoe property or is monotonically monolithic, must X be Gul’ko
compact? V.V. Tkachuk answered this question. He constructed a Corson
compact which fails to be Gul’ko compact and proved that such space has
the Collins-Roscoe property [75].
Also, V.V Tkachuk obtained another explanation for Theorem 11.6. He
proved that compactness in such a result (see Theorem 11.5) can be weakened
to the Lindelöf Σ property.
Theorem 11.8. [78] Suppose that X is a Lindelöf Σ-space and there exists
a weakly σ-point-finite T0-separating family of cozero subsets of X. Then the
space X has the Collins-Roscoe property and, in particular, it is hereditarily
metalindelöf.
The concept of Σs-product was introduced by G. A. Sokolov [65]. Given a
family of spaces {Xt : t ∈ T}, suppose that s = {Tn : n ∈ ω} is a sequence of
subsets of T ; let X =
∏
{Xt : t ∈ T} and fix a point a ∈ X. Given any x ∈ X,
let supp(x) = {t ∈ T : x(t) 6= a(t)} and Ωx = {n ∈ ω : | supp(x) ∩ Tn| < ω}.
Then the set S = {x ∈ X : T =
⋃
{Tn : n ∈ Ωx}} is called the Σs-product
centered at a with respect to the sequence s.
G.A. Sokolov proved the following result:
Theorem 11.9. [65] A compact space X is Gul’ko compact if and only if
X embeds into a Σs-product of real lines.
The following results were proved in [75].
Theorem 11.10. Any Σs-product of compact spaces is a Lindelöf Σ-space.
Proposition 11.11. Any Σs-product S of spaces of countable i-weight has
a weakly σ-point-finite family of cozero sets that T0-separates the points of S.
If S is a Σs-product of compact second countable spaces then by Theorem
11.10 and Proposition 11.11 we have that S is a Lindelöf Σ-space and has a
weakly σ-point-finite family of cozero sets that T0-separates the points of S.
It follows from Theorem 11.8 that S has the Collins-Roscoe property. Since
any second countable space has a second countable compactification, we have
the following consequence (that by Theorem 11.9 is a explanation of Theorem
11.6).
Corollary 11.12. [75] Every Σs-product of second countable spaces has
the Collins-Roscoe property.
The Collins-Roscoe property is also “lifted” by condensations of Lindelöf
Σ-spaces, this makes it possible to generalize Gruenhage’s theorem on Collins-
Roscoe property of Gul’ko compact spaces in the context of their mappings
26 I. PRELIMINARIES
into Σs-products (see Theorem 11.9). Indeed, the following result is a conse-
quence of Proposition 10.7 and Proposition 11.12, since every Σs-product of
spaces of countable i-weight can be condensed into a Σs-product of spaces of
countable weight.
Corollary 11.13. [75] If a Lindelöf Σ-space X condenses into a Σs-
product of spaces of countable i-weight, then X has the Collins-Roscoe prop-
erty.
12. The Collins-Roscoe property in function spaces
G. Gruenhage asked whether the Lindelöf Σ-property of a space X im-
plies that Cp(X) has the Collins-Roscoe property. Although the answer, in
general, is negative (Cp(Dω1
) is a counterexample, see §10). In this section
we will describe a wide class of spaces for which its function spaces have the
Collins-Roscoe property, and in an unexpected form, we also will describe
some class of spaces for which the iterated function spaces have the Collins-
Roscoe property (and hence are hereditarily metalindelöf, see Proposition
10.2).
A. Dow, H. Junnila, J. Pelant proved [27] that if a compact space X
has weight not exceeding ω1 then Cp(X) is hereditarily metalindelöf. Since
every space with the Collins-Roscoe property is metalindelöf, the following
statement (which follows from Corollary 10.5) is a generalization of their
result.
Corollary 12.1. [78] If X is a Lindelöf Σ-space and nw(X) ≤ ω1 then
Cp(X) has the Collins-Roscoe property.
It is well known [72] that if X and Cp(X) are Lindelöf Σ-spaces, then
Cp,n+1(X) and Cp,n(X) are Lindelöf Σ for every n ∈ ω. It was proved in [73]
that in this case Cp,n(X) has a weakly σ-point-finite T0-separating family of
cozero subsets. Then we can apply Theorem 11.8 to see that:
Corollary 12.2. [78] If X and Cp(X) are Lindelöf Σ-spaces then Cp,n(X)
has the Collins-Roscoe property for all n ∈ ω.
If Cp(X) is a Lindelöf Σ-space, since X can be embedded in Cp(Cp(X)); it
follows from Proposition 7.5 that X is monotonically monolithic. This result
can be strengthened to the Collins -Roscoe property. Indeed, if Cp(X) is a
Lindelöf Σ-space it was proved in [49] that υX is a Lindelöf Σ-space and
was proved in [72] that Cp(υX) is Lindelöf Σ, then we can apply Corollary
12.2 to conclude that υX has the Collins-Roscoe property. Finally, since the
Collins-Roscoe property is hereditary we obtain:
Corollary 12.3. [78] If Cp(X) is a Lindelöf Σ-space then X has the
Collins-Roscoe property.
E.A. Reznichenko constructed a Corson compact space B [60] with a
remarkable combination of properties. The space Cp(B) is Lindelöf Σ and
12. THE COLLINS-ROSCOE PROPERTY IN FUNCTION SPACES 27
K-analytic. There is a point b ∈ B such that B is the Stone-Čech compact-
ification of the space Y = B \ {b}. Moreover, the subspace Y = B \ {b} is
pseudocompact and not closed in B. Since Cp(Y ) is a continuous image of
Cp(B), then Cp(Y ) is also a Lindelöf Σ-space. By Corollary 12.3, the space Y
has the Collins-Roscoe property. Thus, Y is an example of a pseudocompact
space with the Collins-Roscoe property which is not compact (see Proposition
9.11).
It was proved in [72] that only the following distributions of the Lindelöf
Σ-property in iterated function spaces are possible:
(1) Cp,n+1(X) are Lindelöf Σ-spaces for all n ∈ ω;
(2) no Cp,n+1(X) is a Lindelöf Σ space for n ∈ ω;
(3) only Cp,n+1(X) with even n ∈ ω are Lindelöf Σ-spaces;
(4) only Cp,n+1(X) with odd n ∈ ω are Lindelöf Σ-spaces.
In particular, for every Gul’ko compact space X we have that Cp,n+1(X)
are Lindelöf Σ-spaces for all n ∈ ω, so Corollary 12.2 and Corollary 12.3
are generalizations of theorem 11.6. Also, from Corollary 12.3 we obtain the
following consequences.
Corollary 12.4. [78] If the space Cp(X) is Lindelöf Σ then the space
Cp,2n(X) has the Collins-Roscoe property for all n ∈ ω.
Corollary 12.5. [78] If Cp(Cp(X)) is a Lindelöf Σ-space, then the space
Cp,2n+1(X) has the Collins-Roscoe property for all n ∈ ω.
Finally, we can see that under certain conditions the fact that Cp(X) is a
Lindelöf Σ-space implies that Cp(X) is a Collins-Roscoe space.
Proposition 12.6. [78] Suppose that X is a space with nw(X) ≤ ω1 and
Cp(X) is a Lindelöf Σ-space. Then Cp(X) has the Collins-Roscoe property.
We know that for any compact space the space Cp(X) is monotonically
monolithic. If X is a Gul’ko compact space, we have that Cp(X) has the
Collins-Roscoe property. In [78], the next question appears: Suppose that X
is a Corson compact space, must the space Cp(X) have the Collins-Roscoe
property?
In [26], A. Dow, H. Junnila, and J. Pelant introduced spaces with point-
finitely expandable networks. Recall that given a space X and a family A
of subsets of X, we say that a family of open sets E = {OA : A ∈ A} is an
open expansion of A if A ⊂ OA for any A ∈ A. A family A of subsets of a
space X is point-finitely expandable if it has a point-finite open expansion in
X. A. Dow, H. Junnila, and J. Pelant proved that if X is a Corson compact
space, then Cp(X) has a σ-point finitely expandable network. Since any σ-
point finitely expandable network is a point-countably expandable network,
by Proposition 11.7, we have that:
Proposition 12.7. [26] For any Corson compact space X the space Cp(X)
has the Collins-Roscoe property.
28 I. PRELIMINARIES
13. Strong monotone monolithicity
Strong monolithic spaces were introduced in [5]. A space is strongly κ-
monolithic if the weight of cl(A) does not exceed κ whenever |A| ≤ κ. A
space is strongly monolithic if it is strongly κ-monolithic for any cardinal κ.
So, it is natural to introduce a strong version of monotone (κ-)monolithicity.
Definition 13.1. [77] Given a set A in a space X we say that a family B
of subsets of X is an external base of A in X if for any x ∈ A and an open
set U ⊂ X with x ∈ U there exists B ∈ B such that x ∈ B ⊂ U .
Definition 13.2. Say that a space X is strongly monotonically monolithic
[77] if, for any A ⊂ X we can assign an external base O(A) to the set cl(A)
in such a way that the following conditions are satisfied:
(1) |O(A)| ≤ max{|A| , ω};
(2) if A ⊂ B ⊂ X then O(A) ⊂ O(B);
(3) if α is an ordinal and we have a family {Aβ : β < α} of subsets of
X such that β < β′ < α implies Aβ ⊂ Aβ′ then O(
⋃
Aβ : β < α) =
⋃
{O(Aβ) : β < α}.
Further, for an infinite cardinal κ, X is said to be strongly monotonically
κ-monolithic [1], if O(A) is defined for all sets A with |A| ≤ κ and satisfies
the above conditions.
It is easy to see that any space with a point countable base is strongly
monotonically monolithic. Every strong monotonically monolithic space is
monotonically monolithic. A cosmic space of uncountable weight is an exam-
ple of a monotonically monolithic space which is not strongly monotonically
monolithic.
The class of strong monotonically monolithic has the following properties.
Proposition 13.3. Strong monotone monolithicity space is preserved by:
(1) countable products [77];
(2) arbitrary subspaces [77];
(3) open and closed functions [44];
(4) open functions with separable fibers [77].
Closed maps do not preserve strong monotonic monolithicity: The space
R is strongly monotonically monolithic being metrizable; the set F = {n :
n ∈ ω} is closed in R. Let Y be the space obtained by collapsing F to a point.
The respective quotient map ϕ : R → Y is closed and it is standard that Y
is not first countable. Therefore, Y is not strongly monotonically monolithic.
An open image of a strongly monotonically monolithic space is not neces-
sarily monolithic: Let Y be the Sorgenfrey line, then Y is not even monolithic.
Since χ(Y ) ≤ ω, there exists a metrizable space X for which there is an open
map f : X → Y . Therefore, X is a strongly monolithic space whose open
continuous image fails to be monolithic.
13. STRONG MONOTONE MONOLITHICITY 29
The following characterization is natural (see Theorem 7.7 and Theorem
9.4).
Theorem 13.4. [75] A space X is strongly monotonically monolithic
(strongly monotonically κ-monolithic) if and only if, for any finite set F ⊂ X,
we can choose a countable family O(F ) of open subsets of X such that, for
every A ⊂ X (with |A| ≤ κ), the family
⋃
{O(F ) : F ∈ [A]<ω} is an external
base for A.
As in the case of monotonically monolithic spaces (see Proposition 9.3),
it is easy to prove that a space X is strongly monotonically monolithic if and
only if it is strongly monotonically κ-monolithic for every infinite cardinal κ.
In fact, the following strong result holds.
Theorem 13.5. [75] A sapce X is strongly monotonically ω-monolithic if
and only if it is strongly monotonically monolithic.
We can also define a strong Collins-Roscoe property, but it is not difficult
to see that such a property is precisely open (G) (see §10). For this reason,
spaces which satisfied open (G) are also called strong Collins-Roscoe spaces
[75]. Then, strong monotone monolithicity is a generalization of open (G).
It is easily seen that any space with a point-countable base B satisfies open
(G). Indeed, the assignment G(x) = {B ∈ B : x ∈ B} witnesses this fact.
Therefore, it is natural to try to extend to strongly monotonically monolithic
spaces the results obtained for the spaces with a point-countable base. The
following results show that sometimes it is possible.
It is now a classic result of Miscenko’s that any compact space with a
point-countable base is metrizable. In fact, it is also well known that this
result holds for countably compact spaces.
Theorem 13.6. [77] If X is a countably compact strongly monotonically
monolithic space then X is metrizable.
On the other hand, there exists a pseudocompact non-metrizable space
X with a point-countable base (see [63]). Therefore, X is a non-metrizable
pseudocompact strongly monotonically monolithic space.
Any space with a point-countable base and caliber ω1 is clearly metrizable.
For strongly monotonically monolithic spaces we have:
Proposition 13.7. [1] Suppose that X is strongly monotonically mono-
lithic space and ω1 is a caliber of X. If additionally d(X) ≤ ω1, then X is
metrizable.
In [77] V.V. Tkachuk asked if a strongly monotonically monolithic space
with caliber ω1 is metrizable. Under the Continuum Hypothesis the answer
is positive. Indeed, under CH, the condition of separability in Proposition
13.7 can be omitted. Observe first that c(X) ≤ ω and hence d(X) ≤ |X| ≤
2χ(X)· c(X) ≤ c = ω1 (see [40]). Thus, we have:
30 I. PRELIMINARIES
Corollary 13.8. [1] Under the Continuum Hypothesis if X is a strongly
monotonically monolithic space and ω1 is a caliber of X, then X is metrizable.
It follows from a general result of J. Chaber [21] that every Lindelöf Σ-
space with a point-countable base is second countable. The following theorem
shows that the assumption on a point-countable base can be weakened to
strong monotonic monolithicity.
Theorem 13.9. [75] If X is a strongly monotonically monolithic Lindelöf
Σ-space, then X is metrizable.
Recall that given a natural number k, a family A of subsets of a space X
is called k-in-countable if every set A ⊂ X with |A| = k is contained in at
most countably many elements of A.
Assume that X is a space and x ∈ X. We say that X is weakly countably
tight at x if there is a countable subset A ofX\{x} such that x is in the closure
of A. A space X is called weakly countably tight if X is weakly countably
tight at every x ∈ X.
Theorem 13.10. [80] Suppose that X is weakly countably tight (or a k-
space) with a k-in-countable base B for some k ∈ N . Then X is strongly
monotonically monolithic.
As we have seen, any space with a point-countable base is a strong Collins-
Roscoe space and any strong Collins-Roscoe space is strongly monotonically
monolithic. In [76] and [77] the following question appears: Is it true that
a space X is strongly monotonically monolithic if and only if X has a point-
countable base? Assuming Martin’s Axiom and ω2 < 2ω, there is a normal
Moore space with a 2-in-finite base that has no point-countable base. This
space is strongly monotonically monolithic by Theorem 13.10, and hence, the
answer to the above question is consistently negative (see [80]).
It is one of the more intriguing open problems in the theory of generalized
metric spaces whether a strong Collins-Roscoe space has a point countable
base, the so called point-countable base problem. Some partial results are
known. V.V. Tkachuk proved in [75] that if X is a Collins-Roscoe space, then
every left-separated subspace Y ⊂ X has a point-countable open expansion
(see §11). Using this fact he proved the following results.
Proposition 13.11. [75]
(1) If X is a Collins-Roscoe space and πχ(X) = ω, then X has a point-
countable π-base.
(2) Every strong Collins-Roscoe space has a point-countable π-base.
(3) If X is a hereditarily Lindelöf strong Collins-Roscoe space, then X
has a point-countable base.
CHAPTER II
Monotone monolithicity
1. Monotone monolithicity of Cp(Cp(X))
It is well known that Cp(Cp(X)) is monolithic if and only ifX is monolithic
(see Theorem I.3.8). Our main goal in this section is to show that a similar
result holds for monotone monolithicity and use it to answer some questions
proposed in [77].
First, we introduce some notations that will be useful.
From now on, we will fix a countable base B(R) for the usual topology in
the set of real numbers R.
Given a space X, for subsets E1, . . . , En of X and subsets U1, . . . , Un of R,
we will use the symbol [E1, . . . , En;U1, . . . , Un] to denote the set {f ∈ Cp(X) :
f(Ei) ⊂ Ui for i = 1, . . . , n}. If E is a family of subsets of X, then W(E)
will be the family of all the sets of the form [E1, . . . , En;B1, . . . , Bn] where
E1, . . . , En ∈ E , B1, . . . , Bn ∈ B(R) and n ∈ ω.
Remarks 1.1. Let X be space, then:
(1) If E is a family of subsets of X, then |W( E)| ≤ max{|E| , ω};
(2) if E and E ′ are families of subsets of X with E ⊂ E ′, then W( E) ⊂
W( E ′);
(3) if {Eβ : β < α} is such that each Eβ is a family of subsets of X and
Eβ ⊂ Eβ′ for β < β′, then W (
⋃
{Eβ : β < α}) =
⋃
{W(Eβ) : β < α}.
Definition 1.2. Let N be a family of subsets of X and let f be a function
from X onto Y . We say that N is a network for Y modulo f if for each x ∈ X
and each open subset U of Y with f(x) ∈ U , there is N ∈ N such that x ∈ N
and f(x) ∈ f(N) ⊂ U .
The following two results show that there exists a duality between the
notions of network modulo a function and external network (see Definition
I.7.1).
Proposition 1.3. Let A ⊂ X. If N is an external network of A in X, then
the family W(N ) of subsets of Cp(X) is a network for πA(Cp(X)) ⊂ Cp(A)
modulo πA.
Proof. Let f ∈ Cp(X) and suppose that πA(f) ∈ U for an open subset U
in πA(Cp(X)). Take x1, . . . , xn ∈ A and B1, . . . , Bn ∈ B(R) such that πA(f) ∈
[x1, . . . , xn;B1, . . . , Bn] ∩ πA(Cp(X)) ⊂ U . Then, there exist N1, . . . , Nn ∈ N
31
32 II. MONOTONE MONOLITHICITY
such that xi ∈ Ni ⊂ f−1(Bi) for i = 1, . . . , n. It is not difficult to see that if
M = [N1, . . . , Nn;B1, . . . , Bn], thenM ∈ W(N ), f ∈M and πA(M) ⊂ U . 
Proposition 1.4. Let f : X → Y be an onto and continuous function
and let N be a family of subsets of X which is a network for Y modulo f .
Then W(N ) is an external network of f ∗(Cp(Y )) in Cp(X).
Proof. Take a continuous function g ◦ f = f ∗(g) ∈ f ∗(Cp(Y )) and
an open subset U of Cp(X) with g ◦ f ∈ U . Take x1, . . . , xn ∈ X and
B1, . . . , Bn ∈ B(R) such that g ◦ f ∈ [x1, . . . , xn;B1, . . . , Bn] ⊂ U . Thus,
g ∈ [f(x1), . . . , f(xn);B1, . . . , Bn]. Because of our hypothesis, we can choose
N1, . . . , Nn ∈ N such that xi ∈ Ni and f(Ni) ⊂ g−1(Bi) for i = 1, . . . , n. So
g ∈ [f(N1), . . . , f(Nn);B1, . . . , Bn]. Hence, if M = [N1, . . . , Nn;B1, . . . , Bn]
then g ◦ f ∈M ⊂ [x1, . . . , xn;B1, . . . , Bn] ⊂ U and M ∈ W(N ). 
In order to prove our main result of this section we need the following
result due to A.V. Arhangel’skii.
Lemma 1.5. [7] Let Y be a dense subspace of the product X =
∏
{Xt :
t ∈ T}, where each space Xt is cosmic. Then, for every continuous real-
valued function f on Y , there exist a countable set S ⊂ T and a continuous
real-valued function g on pS(Y ) such that f = g ◦ pS ↾ Y .
Theorem 1.6. The space Cp(Cp(X)) is monotonically monolithic if and
only if X is monotonically monolithic.
Proof. It is a well known fact that X is homeomorphic to a closed sub-
space of Cp(Cp(X)) (see Proposition I.2.5). Since monotone monolithicity is
a hereditary property, if Cp(Cp(X)) is monotonically monolithic, then X has
this property too.
Now, assume that X is a monotonically monolithic space. For each S ⊂ X
take an external networkO(S) of cl(S) inX in such a way that the assignment
S → O(S) satisfies (1), (2) and (3) in Definition I.7.2.
If f ∈ Cp(Cp(X)), by Lemma 1.5, we can fix a countable set S(f) ⊂ X
and a continuous function g(f) : πS(f)(Cp(X)) → R such that f = g(f)◦πS(f).
For A ⊂ Cp(Cp(X)), we define S(A) =
⋃
{S(f) : f ∈ A}. The operator S
satisfies the following properties:
(1) if A ⊂ Cp(Cp(X)), then |S(A)| ≤ max{|A| , ω};
(2) if A ⊂ B ⊂ Cp(Cp(X)), then S(A) ⊂ S(B);
(3) if {Aβ : β < α} is a family of subsets of Cp(Cp(X)) with Aβ ⊂ Aβ′
when β < β′, then S(
⋃
{Aβ : β < α}) =
⋃
{S(Aβ) : β < α}.
Claim 1. IfA ⊂ Cp(Cp(X)) and S(A) ⊂ S, thenA ⊂ π∗
S(Cp(πS(Cp(X)))).
We will prove this Claim. If f ∈ A, we can take a set S(f) ⊂ X and a con-
tinuous function g(f) : πS(f)(Cp(X)) → R, as before. Let πS the restriction
function from Cp(X) to Cp(S) and let πS
S(f) the restriction function from Cp(S)
to Cp(S(f)). Then, for h(f) = g(f) ◦ πS
S(f) ↾ πS(Cp(X)) : πS(Cp(X)) → R
1. MONOTONE MONOLITHICITY OF Cp(Cp(X)) 33
we have that f = g(f) ◦ πS(f) = h(f) ◦ πS = π∗
S(h(f)) ∈ π∗
S(Cp(πS(Cp(X)))).
This proves the Claim.
Now we are ready to construct a monotonic monolithicity assignment for
the space Cp(Cp(X)). ForA ⊂ Cp(Cp(X)) we takeN (A) = W(W(O(S(A)))).
Because of the above properties of the assignment S, the election of O and
Remarks 1.1, it is easy to verify that N satisfies conditions (1), (2) and (3) in
Definition I.7.2 for the space Cp(Cp(X)). So, in order to prove that Cp(Cp(X))
is monotonically monolithic it is enough to show the following claim.
Claim 2. for any A ⊂ X, N (A) is an external network of cl(A) in
Cp(Cp(X)).
We shall prove this Claim. Let A ⊂ X. Take S = cl(S(A)) ⊂ X and
Y = πS(Cp(X)). By the election of O, we have that O(S(A)) is an ex-
ternal network of S in X. By Proposition 1.3, the family W(O(S(A))) of
subsets of Cp(X) is a network for Y modulo πS. Now, by Proposition 1.4,
W(W(O(S(A)))) = N (A) is an external network of π∗
S(Cp(Y )) in Cp(Cp(X)).
Since S is closed, by Proposition I.2.1 (2), the projection πS is open onto Y
and so πS is a quotient function. By Proposition I.2.3 (2), π∗
S(Cp(Y )) is
closed in Cp(Cp(X)). By Claim 1, we know that A ⊂ π∗
S(Cp(Y )). Thus,
cl(A) ⊂ π∗
S(Cp(Y )). Finally, since N (A) is an external network of π∗
S(Cp(Y ))
in Cp(Cp(X)), N (A) is also an external network of cl(A) in Cp(Cp(X)). 
In the second part of this section we will present some consequences of
our main result.
Corollary 1.7. The following conditions are equivalent:
(1) X is monotonically monolithic;
(2) Cp,2n(X) is monotonically monolithic for some n ∈ ω;
(3) Cp,2n(X) is monotonically monolithic for every n ∈ ω.
The following result gives a complete description of all possible distribu-
tions of the monotone monolithicity property in the spaces Cp,n(X). Notice
that these distributions are very similar to the distributions of the Lindelöf
Σ-property in the spaces Cp,n(X) (See §12 in Chapter I).
Proposition 1.8. One and only one of the following distributions of the
monotone monolithicity property in iterated function spaces happen.
(1) Cp,n(X) is monotonically monolithic for every n ∈ ω;
(2) Cp,n(X) is not monotonically monolithic for every n ∈ ω;
(3) only Cp,n(X) with odd n ∈ ω are monotonically monolithic spaces;
(4) only Cp,n(X) with even n ∈ ω are monotonically monolithic spaces;
Proof. All of the four listed cases exclude themselves mutually. More-
over, these are the only possible distributions because of Corollary 1.7. In
order to prove that all these possible distributions happen, by Corollary 1.7,
it is enough to prove that the following cases must happen: X and Cp(X) are
monotonically monolithic, neither X or Cp(X) are monotonically monolithic,
34 II. MONOTONE MONOLITHICITY
X is not monotonically monolithic but Cp(X) is, and X is monotonically
monolithic but Cp(X) is not.
Now we are going to see that, in fact, all these cases must happen. If
X has a countable network, then X and Cp(X), being cosmic spaces, are
monotonically monolithic; this proves that the first case happens. If S is the
Sorgenfrey line, then d(S) = ω and d(Cp(S)) = iw(S) = ω. Nevertheless,
nw(Cp(S)) = nw(S) > ω. Thus, neither S or Cp(S) are monotonically mono-
lithic; this proves that the second case happens. Consider now the space
[0, ω1]. Since its subspace [0, ω1) does not have the D-property, [0, ω1] is
not hereditary D and then [0, ω1] is not monotonically monolithic (see Theo-
rem I.7.6). Nevertheless, since [0, ω1] is compact, Cp([0, ω1]) is monotonically
monolithic (see Theorem I.6.1), so, the third case happens. Finally, of course,
the last case happens when we take X = Cp([0, ω1]). 
Remark 1.9. Notice that the statements in Theorem 1.6, Corollary 1.7
and Proposition 1.8 remain true if we replace monotone monolithicity by
monotone κ-monolithicity.
It is known that Cp(X) is monotonically monolithic whenever X is a Lin-
delöf Σ-space (see Proposition I.7.5). V.V. Tkachuk asked [77] if the converse
of this result is true, that is: Suppose that Cp(X) is monotonically monolithic,
must X be a Lindelöf Σ-space (Lindelöf space)? He also proposed the fol-
lowing related question: Suppose that the space Cp(Cp(X)) is monotonically
monolithic. Must Cp(X) be a Lindelöf Σ-space?
Now we will use the results obtained in this section to find a counterex-
ample for the second (hence, for the first) question above.
Example 1.10. Let κ be a cardinal number such that κ > ω and let Dκ
the discrete space of cardinality κ. Dκ being metrizable, is monotonically
monolithic, so Cp(Cp(Dκ)) is monotonically monolithic, but Cp(Dκ) = R is
not even normal.
We can also give an example of a Lindelöf space X such that Cp(X) is
not Lindelöf and CpCp(X) is monotonically monolithic. Indeed, let Lκ be
the one-point Lindelöfication of the discrete space Dκ where κ > ω. Lκ
is Lindelöf, and its tightness is not countable. By Asanov’s Theorem (see
Theorem I.4.1 in [3]), Cp(Lκ) is not Lindelöf. But every space with only one
non-isolated point is monotonically monolithic (see Proposition I.7.4). So, Lκ
and Cp(Cp(Lκ)) are monotonically monolithic.
Notice that, as in the last example, in order to construct a non-Lindelöf
space Cp(X) with CpCp(X) monotonically monolithic, is enough to find a
monotonically monolithic space of uncountable tightness.
One of the main characteristics of monotonically monolithic spaces is that
they are hereditary D-spaces. Since monotone monolithicity is preserved
under arbitrary subspaces and closed continuous functions, we have:
2. WHEN IS Cp(X) MONOTONICALLY MONOLITHIC? 35
Corollary 1.11. If X is monotonically monolithic and Y is a closed con-
tinuous image of a subspace of Cp,2n(X) for some n ∈ ω, then Y is hereditarily
D.
Finally the next result show that a for a wide class of spaces monotonic
monolithicity appear in the iterated function spaces (see Chapter I):
Corollary 1.12. If Cp,2n(X) is monotonically monolithic for any n ∈ ω
whenever X belong to one of the following classes: Stratifiable spaces, Collins-
Roscoe spaces (hence spaces with a pint-countably expandable network, spaces
which satisfy open (G), spaces with a point-countable base and metrizable
spaces).
2. When is Cp(X) monotonically monolithic?
In the previous section we saw that Cp(Cp(X)) is monotonically mono-
lithic if and only if X is monotonically monolithic. From this result we can
deduce that there exists a topological property P such that Cp(Cp(X)) is
monotonically monolithic if and only if Cp(X) has P . Indeed, from the proof
of Theorem 1.6, we can see that such a property is given by: Cp(X) has P
if for each A ⊂ X we can assign a countable collection O(A) of subsets of
Cp(X) which is a network for πcl(A)(Cp(X)) modulo πcl(A) and the following
conditions are satisfied:
(1) |O(A)| ≤ max{|A| , ω};
(2) if A ⊂ B ⊂ X then O(A) ⊂ O(B);
(3) if α is an ordinal and we have a family {Aβ : β < α} of subsets of
X such that β < β′ < α implies Aβ ⊂ Aβ′ then O(
⋃
Aβ : β < α) =
⋃
{O(Aβ) : β < α}.
Of course, it is interesting to know if we can define such a property for ev-
ery topological space X, by this reason, in [62] the following general problem
was formulated: Given a class C of topological spaces, determine a topological
property P(C) which satisfies: for every space X ∈ C, Cp(X) is monotonically
monolithic if and only if X has the property P(C). In this section we intro-
duce the notion of monotonically stable space and show that it is the required
property. Indeed, our main result in this section states that for every space
X, the space Cp(X) is monotonically monolithic if and only if the space X is
monotonically stable.
The following facts are easy to verify.
Remark 2.1. Let f be a function from X onto Y .
(1) If a family N of subsets of X is a network for Y modulo f then f(N )
is a network for Y .
(2) If a family N of subsets of X is a network for Y modulo f and g
is a continuous function from Y onto Z then N is a network for Z
modulo g ◦ f .
(3) If N is a network for Y , f−1(N ) is a network for Y modulo f .
36 II. MONOTONE MONOLITHICITY
Recall that given a set X and a family F ⊂ R
X , ∆F : X → R
F denotes
the the diagonal of the family F . The following result will be useful in this
section.
Lemma 2.2. If f is a function from a set X onto a set Y and F ⊂ R
Y ,
then ∆f ∗(F)(X) is homeomorphic to ∆F(Y ).
Proof. Let A = F and B = f ∗(F), both, endowed with the discrete
topology. Since f ∗ : A → B is a bijection then b = (f ∗)∗ : R
B → R
A
is a homeomorphism (see Propositions 2.2 and 2.3 in Chapter I). Since
∆f ∗(F)(X) ⊂ R
B and ∆F(Y ) ⊂ R
A, then, in order to finish the proof,
it suffices to show that b(∆f ∗(F)(X)) = ∆F(X). But this equality follows
from the fact that the function f is surjective and b(∆f ∗(F)(x)) = ∆F(f(x))
for every x ∈ X. Indeed, [b(∆f ∗(F)(x))](g) = [(f ∗)∗(∆f ∗(F)(x))](g) =
[∆f ∗(F)(x) ◦ f ∗](g) = ∆f ∗(F)(x) (f ∗(g)) = ∆f ∗(F)(x)(g ◦ f) = (g ◦ f)(x)
= g(f(x)) = [∆F(f(x))](g) for every g ∈ F . 
Corollary 2.3. If f : X → Y is continuous and onto, then the space
∆f ∗(Cp(Y ))(X) is homeomorphic to ∆Cp(Y )(Y ).
Recall that a space X is stable if and only if for every continuous image Y
of X we have iw(Y ) = nw(Y ). The following result gives a characterization
of stable spaces only in terms of real-valued continuous functions.
Proposition 2.4. A topological space X is stable if and only if for each
subset A ⊂ Cp(X) we can assign a collection N (A) of subsets of X such that
|N (A)| ≤ |A| and N (A) is a network for ∆cl(A)(X) modulo ∆cl(A).
Proof. First, suppose that X is a stable space and take A ⊂ Cp(X).
Let κ = |A|, f = ∆A and Y = f(X). Let Ỹ = Y endowed with the
R-quotient topology generated by f (see §2 in Chapter I). Since Ỹ is a con-
tinuous image of X and the identity function i : Ỹ → Y is a condensation,
we have that nw(Ỹ ) ≤ iw(Ỹ ) ≤ w(Y ) ≤ κ. Notice that A ⊂ f ∗(Cp(Ỹ ));
in fact, if g ∈ A then g = pg ◦ i ◦ f = f ∗(pg ◦ i) ∈ f ∗(Cp(Ỹ )), where
pg : RA → R is the natural projection. Since f : X → Ỹ is a R-quotient
function then f ∗(Cp(Ỹ )) is closed in Cp(X) (see Proposition 2.4 in Chapter
I) and hence cl(A) ⊂ f ∗(Cp(Ỹ )). By Corollary 2.3, ∆f ∗(Cp(Ỹ ))(X) is home-
omorphic to ∆Cp(Ỹ )(Ỹ ) and then ∆f ∗(Cp(Ỹ ))(X) is homeomorphic to Ỹ .
So we have that nw(∆f ∗(Cp(Ỹ ))(X)) ≤ κ. Since ∆cl(A)(X) is a continuous
image of ∆f ∗(Cp(Ỹ ))(X) under the projection from R
f∗(Cp(Y )) to R
cl(A), then
nw(∆cl(A)(X)) ≤ κ. Let N be a network for ∆cl(A)(X) with |N | ≤ κ. Now,
take N (A) = [∆cl(A)(X)]−1(N ). Clearly, |N (A)| ≤ |A| and by Remark 2.1
(3), we have that N (A) is a network for ∆cl(A)(X) modulo ∆cl(A).
Suppose now that X satisfies the second condition in our proposition. Let
Y be a continuous image of X and take a continuous function f from X onto
Y . Let κ = iw(Y ) = d(Cp(Y )) (see Theorem 2.6 in Chapter I). Take a dense
2. WHEN IS Cp(X) MONOTONICALLY MONOLITHIC? 37
subset B of Cp(Y ) of cardinality κ. Let A = f ∗(B). Then f ∗(Cp(Y )) ⊂
cl(A). By hypothesis, there exists a collection N (A) of subsets of X such
that |N (A)| ≤ |A| ≤ |B| = κ and N (A) is a network for ∆cl(A)(X) modulo
∆cl(A). Since ∆f ∗(Cp(Y ))(X) is a continuous image of ∆cl(A)(X) under the
natural projection pf∗(Cp(Y )) from R
cl(A) onto R
f∗(Cp(Y )), by Remark 2.1 (2)
we have that N (A) is a network for ∆f ∗(Cp(Y ))(X) modulo ∆f ∗(Cp(Y )) =
pf∗(Cp(Y )) ◦ ∆cl(A). By Remark 2.1 (1) we have that ∆f ∗(Cp(Y ))(N ) is a
network for ∆f ∗(Cp(Y ))(X) and hence
nw(∆f ∗(Cp(Y ))(X)) ≤ |∆f ∗(Cp(Y ))(N (A))| ≤ |N (A)| ≤ κ.
Finally, since Y is homeomorphic to ∆Cp(Y )(Y ), by Corollary 2.3, Y is
homeomorphic to ∆f ∗(Cp(Y ))(X). Thus, nw(Y ) = nw(∆f ∗(Cp(Y ))(X)) ≤
κ = iw(Y ). This shows that the space X is stable. 
Next, we introduce a property which we will prove is the dual property of
being monotonically monolithic between X and Cp(X).
Definition 2.5. We say that a space X is monotonically stable if for each
A ⊂ Cp(X), we can assign a family O(A) of subsets of X which is a network
for ∆cl(A)(X) modulo ∆cl(A) in such a way that the following conditions
hold:
(1) |O(A)| ≤ max{|A| , ω};
(2) if A ⊂ B ⊂ Cp(X), then O(A) ⊂ O(B);
(3) if {Aα : α < γ} is a family of subsets of Cp(X) with Aα ⊂ Aβ for
α < β, then O(
⋃
{Aα : α < γ}) =
⋃
{O(Aα) : α < γ}.
Further, for an infinite cardinal κ, X is said to be monotonically κ-stable if
O(A) is defined for all sets A with |A| ≤ κ and satisfies the above conditions.
Proposition 2.6. A topological space X is monotonically stable if and
only if for each finite collection F ⊂ Cp(X) we can assign a countable col-
lection N (F ) of subsets of X such that for every A ⊂ Cp(X) the family
⋃
{N (F ) : F ∈ [A]<ω} is a network for ∆cl(A)(X) modulo ∆cl(A).
Proof. First, suppose that for each F ∈ [X]<ω we assign N (F ) which
satisfies the stated conditions. Let O(A) =
⋃
{N (F ) : F ∈ [X]<ω}. It is easy
to check that O satisfies the conditions of the definition of monotonically
stable.
Now we will prove the other implication of our proposition. Suppose X
is monotonically stable, witnessed by operator O. Let N (F ) = O(F ) for
every finite set F ⊂ Cp(X). We will show that
⋃
{N (F ) : F ∈ [A]<ω} is a
network for ∆cl(A)(X) modulo ∆cl(A) for every A ⊂ Cp(X). To this end, fix
A ⊂ Cp(X) and let Y = f(X) where f = ∆cl(A). Let x ∈ X and U an open
subset of Y with f(x) ∈ U . Then x ∈ N and f(N) ⊂ U for some N ∈ O(A).
Let F ⊂ A have minimal cardinality such that N ∈ O(F ). We claim that F
is finite. Suppose otherwise, and let F = {xα : α < κ} where κ = |F |. Now
let Fα = {xβ : β < α}. Since F =
⋃
{Fα : α < κ} then, by condition 3) in
38 II. MONOTONE MONOLITHICITY
the definition of monotonically stable, we have O(F ) =
⋃
{O(Fα) : α < κ};
so N ∈ O(Fα) for some α < κ. But |Fα| < |F |; this is a contradiction. Thus,
F ∈ [A]<ω and N ∈ O(F ) = N (F ). 
Now we will prove one of the main results in this section.
Let X and Y be topological spaces. Suppose that N (F ) is a family of
subsets of Y for each F ∈ [X]<ω. We define the assignment N ′ as fol-
lows: N ′(F ) =
⋃
{N (E) : E ∈ [F ]<ω} for each F ∈ [X]<ω. Notice that
⋃
{W(N ′(F )) : F ∈ [A]<ω} = W(
⋃
{N ′(F ) : F ∈ [A]<ω}) for every A ⊂ X.
Theorem 2.7. For every space X, the space Cp(X) is monotonically
monolithic if and only if the space X is monotonically stable.
Proof. First suppose that X is a monotonically stable space and take
an assignment N that witnesses this fact as in Proposition 2.6. For each
finite subset F of Cp(X), take O(F ) = W(N ′(F )). Notice that O(F ) is
countable. To see that Cp(X) is monotonically monolithic we show that the
operator O satisfies the conditions in Proposition 7.7 in Chapter I. Indeed,
take A ⊂ Cp(X). Let f = ∆cl(A) and Y = f(X). By the election of N , the
family
⋃
{N ′(F ) : F ∈ [A]<ω} is a network for Y modulo f . By Proposition
1.4, we have that
⋃
{W(N ′(F )) : F ∈ [A]<ω} = W(
⋃
{N ′(F ) : F ∈ [A]<ω}) is
an external network for f ∗(Cp(Y )) in Cp(X). Notice that cl(A) ⊂ f ∗(Cp(Y )).
In fact, if g ∈ cl(A) then g = p ◦ f = f ∗(p) ∈ f ∗(Cp(Y )), where p = pg ↾ Y
is the restriction of the projection pg : Rcl(A) → R. Thus,
⋃
{O(F ) : F ∈
[A]<ω} =
⋃
{W(N ′(F )) : F ∈ [A]<ω} is an external network for cl(A) in
Cp(X).
Suppose now that Cp(X) is a monotonically monolithic space with its re-
spective operator O as in Proposition 7.7 in Chapter I. Let ψ = ∆Cp(X) be
the canonical embedding of X in Cp(Cp(X)) (see §2 in Chapter I). For each
finite set F ⊂ X, let N (F ) be equal to ψ−1(W(O′(F ))). We will show that
the operator N satisfies the conditions in Proposition 2.6 for the space X.
Clearly N (F ) is countable. Fix A ⊂ Cp(X). Since
⋃
{O′(F ) : F ∈ [A]<ω} is
an external network for cl(A) then, by Proposition 1.3,
⋃
{W(O′(F )) : F ∈
[A]<ω} = W(
⋃
{O′(F ) : F ∈ [A]<ω}) is a network for πcl(A)(Cp(Cp(X))) mod-
ulo πcl(A). Let f = ∆cl(A) and Y = f(X). Notice that f = pcl(A) ◦ ψ where
pcl(A) is the projection from R
Cp(X) onto R
cl(A). Since ψ(X) ⊂ Cp(Cp(X))
then Y = f(X) = pcl(A)(ψ(X)) ⊂ pcl(A)(Cp(Cp(X))) = πcl(A)(Cp(Cp(X))).
We shall verify that
⋃
{N (F ) : F ∈ [A]<ω} is a network for Y modulo f .
Take x ∈ X such that f(x) ∈ U for some open set U ⊂ Y . Let V be an open
set in πcl(A)(Cp(Cp(X))) such that V ∩Y = U . Since πcl(A)(ψ(x)) = f(x) ∈ V
and
⋃
{W(O′(F )) : F ∈ [A]<ω} is a network for πcl(A)(Cp(Cp(X))) modulo
πcl(A), there exist E ∈ [A]<ω and N ∈ W(O′(E)) such that ψ(x) ∈ N and
πcl(A)(ψ(x)) ∈ πcl(A)(N) ⊂ V . Thus, for M = ψ−1(N) ∈ ψ−1(W(O′(E))) =
N (E) ⊂
⋃
{N (F ) : F ∈ [A]<ω}, we have that x ∈ M and f(x) ∈ f(M) =
πcl(A)(ψ(M)) ⊂ πcl(A)(N) ∩ Y ⊂ V ∩ Y = U . 
2. WHEN IS Cp(X) MONOTONICALLY MONOLITHIC? 39
As we saw in Chapter I, Cp(X) is monolithic if and only if X is stable
and X is monolithic if and only if Cp(X) is stable. By Theorems 2.7 and
1.6, we obtain the following two corollaries which show that the duality be-
tween monolithic and stable spaces hold for monotonically monolithic and
monotonically stable spaces.
Corollary 2.8. For every space X the space Cp(X) is monotonically stable
if and only if the space X is monotonically monolithic.
Proof. Cp(X) is monotonically stable if and only if Cp(Cp(X)) is mono-
tonically monolithic if and only if X is monotonically monolithic. 
Corollary 2.9. The space Cp(Cp(X)) is monotonically stable if and only
if X is monotonically stable.
Remark 2.10. Naturally, statements in Theorem 1.6, Theorem 2.7, Corol-
lary 2.8 and Corollary 2.9 remain true if we replace monotonically monolithic
and monotonically stable by monotonically κ-monolithic and monotonically
κ-stable, respectively.
Now we preve basic properties of monotonically stable spaces.
Proposition 2.11. Let f : X → Y be a continuous function from a space
X onto a space Y . If X is monotonically stable, then Y is monotonically
stable.
Proof. By Theorem 2.7, Cp(X) is monotonically monolithic. Since the
space Cp(Y ) is homeomorphic to f ∗(Cp(Y )) and monotone monolithicity is
hereditary, then Cp(Y ) shares this property. Finally, by Theorem 2.7, Y is
monotonically stable. 
Proposition 2.12. A space which is the union of a countable set of mono-
tonically stable subspaces is monotonically stable.
Proof. Let Y =
⋃
{Yn : n ∈ ω} where each Yn is a monotonically stable
space. Then Cp(Yn) is monotonically stable for every n ∈ ω. Let us consider
the free topological sum X = ⊕{Yn : n ∈ ω} of the family {Yn : n ∈
ω}. By Theorem 7.3 in Chapter I, the space Cp(X) =
∏
{Cp(Yn) : n ∈ ω}
is monotonically monolithic, so X is monotonically stable. Clearly, Y is a
continuous image of X, then Y is monotonically stable. 
Proposition 2.13. Let κ a cardinal. If Cp(Xα) is monotonically stable
for any α < κ, then the space
∏
{Cp(Xα) : α < κ} is monotonically stable.
Proof. First, notice that Xα is monotonically monolithic for every α <
κ. Let us consider the free topological sum X = ⊕{Xα : α < κ}. It is not dif-
ficult to verify that the union of a locally finite family of closed monotonically
monolithic subsets of a space is monotonically monolithic. In particular, X is
monolithic. Thus Cp(X) =
∏
{Cp(Xα) : α < κ} is monotonically stable. 
40 II. MONOTONE MONOLITHICITY
Corollary 2.14. If a space Cp(X) is monotonically stable, then the space
(Cp(X))κ is monotonically stable for every cardinal κ.
It is easy to deduce from Proposition 2.4 that any monotonically stable
space is stable. By Proposition 7.5 in Chapter I and Theorem 2.7, Lindelöf Σ-
spaces and in particular compact spaces are monotonically stable. As we have
seen, [0, ω1] is not monotonically monolithic, so Cp([0, ω1]) is a stable space
which is not monotonically stable. Moreover, if X is the subspace of [0, ω2)
of all ordinals of uncountable cofinality, X is monotonically ω-monolithic but
not monotonically ω1-monolithic [1], so Cp(X) is monotonically ω-stable but
not monotonically ω1-stable. A discrete space D of cardinality c is not even
ω-stable, but is homeomorphic to a closed subspace of Rc. Thus, monotone
stability is not inherited, in general, by closed subspaces and uncountable
unions.
3. Monotone monolithicity of Cp(X) when X is a Σ-product
In Example 1.10 we saw that Cp(Cp(Lκ)) is monotonically monolithic,
where Lκ is the one-point Lindelöfication of the discrete space of cardinality
κ. It is well known that Cp(Lκ) is homeomorphic to the Σ-product in a
product of κ copies of the real line. Also, it is known that Cp(X) is monolithic
when X is a Σ-product in a product of cosmic spaces and even in a product
of Lindelöf Σ-spaces (see [3]). So, is natural to ask if these results hold for
monotone monolithicity.
In this section we show that, indeed, in the case of products of cosmic
spaces the answer is positive by showing a more general result. We use such a
result to give another proof of the fact that Cp(X) is monotonically monolithic
when X is a compact space (see Theorem 6.1 in Chapter I). On the other
hand, we can only prove that Cp(X) is monotonically monolithic when X is
the product of a family of Lindelöf Σ-spaces.
The following technical result is well known and easy to prove; we present
the proof for completeness.
Lemma 3.1. Let X =
∏
{Xα : α < κ} be a topological product, F a
subset of X, S a subset of κ and f : F → R a continuous function. If
pS ↾ F : F → pS(F ) is a quotient function and f(x) = f(x′) whenever
x, x′ ∈ F and pS(x) = pS(x
′), then there exists g : πS(F ) → R, continuous,
such that f = g ◦ pS ↾ F .
Proof. We define g : pS(F ) → R as follows: if y = pS(x) ∈ pS(F ) for
some x ∈ F , we take g(y) = f(x). Since f(x) = f(x′) for each pair x, x′ ∈ X
with pS(x) = pS(x
′), g is well defined. In order to prove that g is continuous,
observe that for an open subset U of R, the continuity of f implies that the
set f−1(U) = (pS ↾ F )−1(g−1(U)) is open in F . Since pS ↾ F is a quotient
function, then g−1(U) is open in pS(F ). Because of the definition of g, we
have f = g ◦ pS ↾ F . 
3. MONOTONE MONOLITHICITY OF Cp(X) WHEN X IS A Σ-PRODUCT 41
Now we are ready to prove one of our main results in this section.
Theorem 3.2. Let X =
∏
{Xα : α < κ} be a product of cosmic spaces
and let Y be a dense subset of X. Suppose that F is a C-embedded subspace
of Y such that for each S ⊂ κ the function qS = pS ↾ F : F → pS(F ) is a
quotient function. Then Cp(F ) is monotonically monolithic.
Proof. For each f ∈ Cp(F ), since F is a C-embedded subspace of Y , we
can fix an extension f̃ ∈ Cp(Y ) of f . By Lemma 1.5, since Y is dense inX and
f̃ : Y → R is a continuous function, we can fix a countable set S(f) ⊂ κ, and
a continuous function gS(f) : πS(f)(Y ) → R such that f̃(x) = gS(f) ◦ pS(f) ↾ Y .
If A ⊂ Cp(Y ), let S(A) =
⋃
{S(f) : f ∈ A}. Observe that operator S has
the following properties:
(1) if A ⊂ Cp(Y ), then |S(A)| ≤ max{|A| , ω};
(2) if A ⊂ B ⊂ Cp(Y ), then S(A) ⊂ S(B);
(3) if {Aα : α < γ} is a family of subsets of Cp(Y ) with Aα ⊂ Aβ when
α < β < γ, then S(
⋃
{Aα : α < γ}) =
⋃
{S(Aα) : α < γ}.
For each α < κ, let N (Xα) be a countable network of Xα. For each
S ⊂ κ we will denote by NF (S) the collection of all subsets of X of the
form
∏
{Nα : α < κ} ∩ F where Nα ∈ N (Xα) if α ∈ T and Nα = Xα for
α ∈ κ\T , where T is a finite subset of S. Observe that NF (S) is a network for
qS(F ) modulo qS for each S ⊂ κ, and that the operator NF has the following
properties:
(1) if S ⊂ κ, then |NF (S)| ≤ max{|S| , ω};
(2) if S ⊂ S ′ ⊂ κ, then NF (S) ⊂ NF (S
′);
(3) if {Sα : α < γ} is a family of subsets of κ with Sα ⊂ Sβ when α < β,
then NF (
⋃
{Sα : α < γ}) =
⋃
{NF (Sα) : α < γ}.
Now we are ready to construct a monotonic monolithicity operator in
Cp(F ). If A ⊂ Cp(F ), let O(A) = W(NF (S(A))). Because of Remark 1.1
and the above properties of the operators S and NF , it is easy to verify that
the operator O satisfies conditions (1), (2) and (3) in Definition 7.2. So, in
order to finish the proof it is enough to prove:
Claim. O(A) is an external network of cl(A) in Cp(F ) for every A ⊂
Cp(F ).
We shall prove the Claim. Since NF (S(A)) is a network for qS(A)(F )
modulo qS(A), the family O(A) = W(NF (S(A))) is an external network for
q∗
S(A)(Cp(qS(A)(F ))) in Cp(F ) (see Proposition 1.4). Thus, in order to prove
that O(A) is an external network of cl(A) in Cp(F ), it is enough to show that
cl(A) ⊂ q∗
S(A)(Cp(qS(A)(F ))).
Let g ∈ cl(A). Take x, x′ ∈ F such that qS(A)(x) = qS(A)(x
′). For each
f ∈ A, because of the definition of S(A) we have that pS(f)(x) = pS(f)(x
′)
and so f(x) = f̃(x) = gS(f) ◦ pS(f)(x) = gS(f) ◦ pS(f)(x
′) = f̃(x′) = f(x′).
If g(x) 6= g(x′), we can choose disjoint open subsets B,B′ ∈ B(R) with
g(x) ∈ B and g(x′) ∈ B′. Then, the open set U = {f ∈ Cp(F ) : f(x) ∈
42 II. MONOTONE MONOLITHICITY
B and f(x′) ∈ B′} contains g and has an empty intersection with A; which
is not possible since g ∈ cl(A). Thus g(x) = g(x′). Since the projection
qS(A) : F → qS(A)(F ) is a quotient map, by Lemma 3.1, there is a continuous
function gS(A) : qS(A)(F ) → R such that g = gS(A) ◦ qS(A) = q∗
S(A)(gS(A)) ∈
q∗
S(A)(Cp(qS(A)(F ))). Therefore, cl(A) ⊂ q∗
S(A)(Cp(qS(A)(F ))). 
Remark 3.3. Let us observe that if in the above result we only have
that for each S ⊂ κ with cardinality at most λ, the function qS = pS ↾
F : F → pS(F ) is a quotient function, then we can conclude that Cp(F ) is
monotonically λ-monolithic.
Suppose that η = {Xt : t ∈ T} is a family of topological spaces, X =
∏
{Xt : t ∈ T} is the topological product of the family η, x∗ is a point in X
and κ is a cardinal number. Then the Σκ-product of η with basic point x∗ is
the subspace of X consisting of all points x ∈ X such that only less than κ
coordinates x(t) of x are distinct from the corresponding coordinates x∗(t) of
x∗. This subspace is denoted by Σκ{Xt : t ∈ T} or by Σκη. Sometimes Σκη
will be called the Σκ-product of η at x∗.
Let us observe that given a family η of topological spaces, then Ση = Σω1
η
and ση = Σωη.
We know that if X =
∏
{Xt : t ∈ T} is the topological product of the
family
∏
{Xt : t ∈ T}, x∗ is a point in X, and κ is a cardinal number; then
for Y = Σκ{Xt : t ∈ T} and every S ⊂ T the function pS ↾ Y : Y → pS(Y )
is open and hence is a quotient function. So we can apply Theorem 3.2 to
obtain:
Corollary 3.4. Let κ be an infinite cardinal number and Y a Σκ-product
of a family of cosmic spaces, then Cp(Y ) is monotonically monolithic.
Corollary 3.5. Let κ be an infinite cardinal number and Y a Σκ-product
of a family of cosmic spaces, then Y is monotonically stable.
Corollary 3.6. If X is a σ-product of a family of spaces, each of them
with countable network, then Cp,n(X) is monotonically monolithic for every
n ∈ ω.
Proof. Since every space with countable network is monotonically mono-
lithic, by Theorem I.7.3 (2), we have that X is monotonically monolithic.
Because of Corollary 3.4, the space Cp(X) is monotonically monolithic. By
Corollary 1.8, it happens that Cp,n(X) is monotonically monolithic for any
n ∈ ω. 
Now we give another proof of Theorem 6.1 for compact spaces.
Corollary 3.7. Cp(X) is monotonically monolithic for any compact space
X.
Proof. Let X be a compact space. Take an embedding f : X → R
κ for
some cardinal κ. Take F = f(X) and Y = R
κ. Clearly, pS ↾ F : F → pS(F )
3. MONOTONE MONOLITHICITY OF Cp(X) WHEN X IS A Σ-PRODUCT 43
being closed is a quotient function, for any S ⊂ κ. By Theorem 3.2, Cp(F ) is
monotonically monolithic. Thus Cp(X) is monotonically monolithic. 
Lindelöf Σ-spaces are a natural generalization of cosmic spaces. Indeed,
cosmic spaces are exactly the Lindelöf σ-spaces, so it is interesting to know
when the results obtained until now hold for Lindelöf Σ-spaces. We will work
in this direction.
We will need the following results.
Lemma 3.8. [3] Let X =
∏
{Xα : α < κ} be a topological product,
T = σ
∏
{Xα : α < κ}, Y a subspace of X which contains T and λ an infinite
cardinal. Assume that for each finite set K ⊂ κ, l(
∏
{Xα : α ∈ K}) ≤ λ. If
f : Y → Z is a continuous and iw(Z) ≤ λ, then there is a set S ⊂ κ with
|S| ≤ λ such that if x, x′ ∈ Y and pS(x) = pS(x
′) then f(x) = f(x′).
Lemma 3.9. [29] Let X =
∏
{Xα : α < κ} be a product of topological
spaces. Let Kα be a compact subset of Xα for each α < κ. If U is an open
subset of X such that K =
∏
{Kα : α < κ} ⊂ U , then there exist a finite set
S ⊂ κ and a family {Uα : α ∈ S} with Kα ⊂ Uα ⊂ Xα for each α ∈ S such
that K ⊂
⋂
{p−1
α (Uα) : α ∈ S} ⊂ U .
The following result gives a generalization of Proposition 7.5 in Chapter
I.
Theorem 3.10. Let X =
∏
{Xα : α < κ} be a product of Lindelöf Σ-
spaces. Then, Cp(X) is monotonically monolithic.
Proof. For each α < κ, let Kα be a compact cover of Xα and let Nα be
a countable network for Xα modulo Kα. For S ⊂ κ, we denote by NS the
family of all subsets of X of the form
⋂
{p−1
α (Nα) : α ∈ S0} where Nα ∈ Nα
for each α ∈ S0 and S0 is a finite subset of S. Note that |NS| ≤ max{|S| , ω};
in particular, if S is countable then NS is countable. For each family E of
subsets of X, let F(E) be the collection {
⋃
A : A ⊂ E and |A| < ω} and
CS(E) = {N \ E : N ∈ NS, E ∈ E}.
Let f : X → R be a continuous function. The product of a finite collection
of Lindelöf Σ-spaces is a Lindelöf Σ-space. Moreover, iw(R) = ω. So, by
Lemma 3.8, we can fix a countable set S(f) ⊂ κ such that if x, x′ ∈ X and
pS(x) = pS(x
′) then f(x) = f(x′). Since the projection pS : X → XS =
∏
{Xα : α ∈ S} is open for each S ⊂ κ, there exists a continuous function
g(f) : XS(f) → R such that f = g(f) ◦ pS(f) (see Proposition 4.2). For a
finite set F ⊂ Cp(X), let S(F ) =
⋃
{S(g) : g ∈ F}, EF = {g−1(B) : g ∈
F,B ∈ B(R)} and N (F ) = W(CS(F )(F(EF ))). Observe that the family N (F )
is countable.
In order to prove that Cp(X) is monotonically monolithic, it is enough
to prove, because of Proposition I.7.7, that if A ⊂ Cp(X) then
⋃
{N (F ) :
F ∈ [A]<ω} is an external network of cl(A) in Cp(X). Take A ⊂ Cp(X) and
f ∈ cl(A).
44 II. MONOTONE MONOLITHICITY
Claim. If x ∈ X, B ∈ B(R) and f(x) ∈ B, then there are F ⊂ A, finite,
and P ∈ CS(F )(F(EF )) such that x ∈ P and f ∈ [P ;B].
We are going to prove the Claim. For each α < κ, let Kα ∈ Kα with
x(α) ∈ Kα. Thus, x ∈ K =
∏
{Kα : α < κ}. For every y ∈ K \ f−1(B),
take a By ∈ B(R) such that f(y) ∈ By and f(x) 6∈ cl(By), and take a
function gy ∈ A such that gy(x) ∈ B \ cl(By) and gy(y) ∈ By. The family
{g−1
y (By) : y ∈ K \ f−1(B)}∪ {f−1(B)} covers K. Since K is compact, there
is a finite subset T0 of K \ f−1(B) such that {g−1
y (By) : y ∈ T0} ∪ {f−1(B)}
covers K. If U =
⋃
{g−1
y (By) : y ∈ T0} ∪ f
−1(B), then K ⊂ U . By Lemma
3.9, there are a finite subset S0 ⊂ κ and a family {Uα : α ∈ S0} of open
subsets with Kα ⊂ Uα ⊂ Xα for each α ∈ S0, such that K ⊂
⋂
{p−1
α (Uα) :
α ∈ S0} ⊂ U . Let S(A) =
⋃
{S(g) : g ∈ A} and S = S0 ∩ S(A). Hence,
K ⊂
⋂
{p−1
α (Uα) : α ∈ S} ⊂ U . In fact, this follows from the fact that for
each g ∈ A, g ↾ E is constant for every set E of the form p−1
S(A)(pS(A)(x)).
Thus, each h ∈ cl(A) satisfies this property too. For each α ∈ S, take
gα ∈ A with a ∈ S(gα). Since for each α ∈ S we have that Kα ∈ Kα,
Kα ⊂ Uα and Nα is a network for Xα modulo Kα, there exists Nα ∈ Nα
with Kα ⊂ Nα ⊂ Uα. Let N =
⋂
{p−1
α (Nα) : α ∈ S}. Then K ⊂ N ⊂ U .
Finally, let P = N \
⋃
{g−1
y (By) : y ∈ T0}, then x ∈ P , P ∈ CS(F )(F(EF )) for
F = {gy : y ∈ T0} ∪ {gα : α ∈ S} ⊂ A and f ∈ [P ;B]. This ends the proof
of the Claim.
Now suppose that f ∈ U for an open subset U of Cp(X). Take x1, . . . , xn ∈
X and B1, . . . , Bn ∈ B(R) such that f ∈ [x1, . . . , xn;B1, . . . , Bn] ⊂ U . Be-
cause of the Claim, for each i = 1, . . . , n there are Fi ⊂ A, finite, and
Pi ∈ CS(Fi)(F(EFi
)) such that xi ∈ Pi and f ∈ [Pi ;Bi]. Take F =
⋃
{Fi :
i = 1, . . . , n} ⊂ A; then Pi ∈ CS(F )(F(EF )) for each i = 1, . . . , n. Finally,
if M = [P1, . . . , Pn;B1, . . . , Bn], then f ∈ M ⊂ [x1, . . . , xn;B1, . . . , Bn] ⊂ U ,
where M ∈ W(CS(F )(F(EF ))) = N (F ). Therefore,
⋃
{N (F ) : F ∈ [A]<ω} is
an external network of cl(A) in Cp(X). 
Corollary 3.11. Any product of Lindelöf Σ-spaces is monotonically stable.
4. Monotone monolithicity of Cp(X) and cozero sets
In [17] R. Buzyakova investigated how the Lindelöf property of the func-
tion space Cp(X, Y ) is influenced by slight changes in X or Y . In particular,
it is interesting to know what happens to Cp(X) if we remove one point from
X. It is known that removing a point of countable tightness from a zero-
dimensional compact space may destroy the Lindelöf property of a function
space. For example, let X = D ∪ {∞} be the one-point compactification of
an uncountable discrete space D. It is known that X is an Eberlein compact
space (see [3]). Therefore, Cp(X) is Lindelöf. The space Cp(D) is not Lin-
delöf because it is homeomorphic to R
D. However, R. Buzyakova proved that
removing a point of countable character from a zero-dimensional compact
4. MONOTONE MONOLITHICITY OF Cp(X) AND COZERO SETS 45
space does not affect the Lindelöf property of the function space. She asked
if the same holds for every compact space, or for any space X.
O.G. Okunev [48] proved that if Y is a cozero set in X, then Cp(Y ) is
a continuous image of a closed subset of Cp(X)ω. Okunev’s result gener-
alizes the result of R. Buzyakova. Indeed, he also proved that if Cp(X) is
Lindelöf, then Cp(X)ω shares this property whenever X is a σ-compact zero-
dimensional space. Thus, removing even a zero set from a zero-dimensional
σ-compact space does not affect the Lindelöf property of the function space.
In this section we will improve the first of the above of Okunev’s results
by showing that: if Y is a cozero set in X, then Cp(Y ) is homeomorphic to a
closed subset of Cp(X)ω. As a consequence we have that if Cp(X) is monoton-
ically monolithic and Y is a cozero subset of X, then Cp(Y ) is monotonically
monolithic.
Theorem 4.1. Let X be a space and let Y be a cozero subset of X. Then,
Cp(Y ) is homeomorphic to a closed subspace of Cp(X)ω.
Proof. We began the proof of our theorem following Theorem 2.1 in [48].
Let h : X → [0, 1] be a continuous function such that Y = h−1((0, 1]). For
each n ∈ ω, define Fn = h−1([1/(n+1), 1]) and F = X \Y . Of course, F and
Fn, n ∈ ω, are zero sets. Note that Fn ⊂ int(Fn+1) and Y =
⋃
{Fn : n ∈ ω}.
Consider the closed subset Z = {f ∈ Cp(X) : f(F ) ⊂ {0}} of Cp(X).
Consider the set
P = {G ∈ Zω : G(n) ↾ Fn = G(m) ↾ Fn for each n,m ∈ ω,m ≥ n} .
Then
P =
⋂
n∈ω
⋂
m≥n
⋂
x∈Fn
{G ∈ Zω : G(n)(x) = G(m)(x)} .
So, P is closed in Zω and hence in Cp(X)ω. Now, we are going to prove that
Cp(Y ) is a continuous image of P . Take the function T : P → R
Y defined as
T (G)(x) = G(n)(x) if x ∈ Fn.
It is clear that T is well defined. Let G ∈ P and x ∈ Y . Thus, x ∈
Fn ⊂ int(Fn+1) for some n ∈ ω. Since T (G) ↾ Fn+1 = G(n + 1) ↾ Fn+1,
T (G) coincides with the continuous function G(n + 1) in Fn+1 which is a
neighborhood of x. Therefore, T (G) is continuous at x. So, we obtain that
T (G) is continuous on Y , and so T (P ) ∈ Cp(Y ).
Now, we are going to show that Cp(Y ) ⊂ T (P ). Fix a continuous function
θ : [0, 1] → [0, 1] such that θ(1) = 1 and θ([0, 1/2]) = {0}. Since F and Fn
are zero sets for every n ∈ ω, we can also fix, for each n ∈ ω, a continuous
function hn : X → [0, 1] such that hn(F ) ⊂ {0} and hn(Fn) ⊂ {1}. Take
sn = θ ◦ hn. Then sn : X → [0, 1] is continuous, sn(Fn) = {1} and sn is
equal to 0 in some neighborhood of F . If f ∈ Cp(Y ), then the function
46 II. MONOTONE MONOLITHICITY
g(f, n) : X → R defined by the rule
g(f, n)(x) =
{
f(x)sn(x) if x ∈ Y ;
0 if x ∈ F,
is continuous on X and coincides with f on Fn. Note that if Gf ∈ Cp(X)ω
is defined as Gf (n) = g(f, n) for each n ∈ ω, then Gf ∈ P and T (Gf ) = f .
This concludes the proof that T (P ) = Cp(Y ). Now, we are going to prove
that T is a continuous function. Take [x;B] in Cp(Y ) where x ∈ Y and
B ∈ B(R). Let m ∈ ω such that x ∈ Fm. Hence x ∈ Fn for each n ≥ m; so
G(n)(x) = G(m)(x) for each G ∈ P and n ≥ m. Therefore, the set
T−1([x,B]) = {G ∈ P : G(m)(x) ∈ B} = P ∩ {H ∈ Cp(X)ω : H(m)(x) ∈ B}
is an open subset of P . Thus Cp(Y ) is a continuous image of the subspace P
of Cp(X)ω.
Until here, we follow Okunev’s proof. Now, we are going to go a little
further, we are going to prove that, in fact, Cp(Y ) is homeomorphic to a
closed subset of Cp(X)ω.
For each n ∈ ω, consider the function Sn : Cp(Y ) → Cp(X) given by
Sn(f) = g(f, n) for each f ∈ Cp(Y ). We now prove that Sn is a continuous
function. Let A ⊂ Cp(Y ) and f ∈ clCp(Y )(A). Let W = [Sn(f);K; ǫ] = {t ∈
Cp(X) : |Sn(f)(x)− t(x)| < ǫ, for each x ∈ K} be a canonical neighborhood
of Sn(f), where K ⊂ X is finite and ǫ > 0. Take K0 = K ∩ Y and r ∈
[f ;K0; ǫ] ∩ A. If x ∈ K ∩ F , then |Sn(r)(x)− Sn(f)(x)| = 0 < ǫ. If x ∈
K ∩ Y = K0, then
|Sn(r)(x)− Sn(f)(x)| = |r(x)sn(x)− f(x)sn(x)| ≤ |r(x)− f(x)| < ǫ.
This proves that Sn(r) ∈ W ∩ Sn(A). Thus, Sn(f) ∈ clCp(X)(Sn(A)). So,
we have proved that each Sn is continuous. Take the diagonal of the family
of functions {Sn : n ∈ ω}: S = ∆{Sn : n ∈ ω} : Cp(Y ) → Cp(X)ω. Then,
S is a continuous function. Moreover, S(f) = Gf ; so, S(Cp(Y )) ⊂ P and
T (S(f)) = f .
Since T (S(f)) = f for each f ∈ Cp(Y ), S is injective. Hence, from the
continuity of S and T we can deduce that S is a homeomorphism from Cp(Y )
to the subspace S(Cp(Y )) of Cp(X)ω.
Finally we are going to prove that S(Cp(Y )) is a closed subset of Cp(X)ω.
Since P is closed in Cp(X)ω, it is enough to prove that S(Cp(Y )) is closed in
P . We will first prove that if G ∈ P and, if for each n,m ∈ ω with n < m,
and x ∈ Fm we have G(n)(x) = G(m)(x)sn(x), then G ∈ S(Cp(Y )). It is
sufficient to prove that G = S(T (G)). Let n ∈ ω and x ∈ X. If x ∈ F then
G(n)(x) = 0 = S(T (G))(n)(x). If x ∈ Y , we can choose m ∈ ω such that
n < m and x ∈ Fm. Hence
G(n)(x) = G(m)(x)sn(x) = T (G)(x)sn(x) = S(T (G))(n)(x).
5. MONOTONE κ-MONOLITHICITY AND FUNCTION SPACES 47
This concludes the proof of the equality G = S(T (G)). On the other hand, if
G = Gf ∈ S(Cp(Y )) for some f ∈ Cp(Y ), then for each n,m ∈ ω with n < m
and x ∈ Fm we have G(n)(x) = g(f, n)(x) = f(x)sn(x) = G(m)(x)sn(x).
Therefore, G ∈ S(Cp(Y )) if and only if G ∈ P and G(n)(x) = G(m)(x)sn(x)
for each n,m ∈ ω with n < m and x ∈ Fm; that is,
S(Cp(Y )) = P ∩
⋂
m∈ω
⋂
n<m
⋂
x∈Fm
{G ∈ Cp(X)ω : G(n)(x) = G(m)(x)sn(x)} .
So, S(Cp(Y )) is closed in P . 
Corollary 4.2. Let P be a class of spaces which is invariant with respect
to countable powers and closed subsets. If Cp(X) ∈ P and Y is a cozero
subset of X, then Cp(Y ) ∈ P.
Since every open Fσ subset of a normal space is a cozero set, we obtain:
Corollary 4.3. Let P be a class of spaces invariant with respect to count-
able powers and closed subsets. If X is normal, Cp(X) ∈ P and Y is an Fσ
open subset of X, then Cp(Y ) ∈ P.
As we saw in Chapter I, the class of monotonically monolithic spaces and
the class of Collins-Roscoe spaces are invariant with respect to countable
powers and arbitrary subsets. Also, X is monotonically stable if and only if
Cp(X) is monotonically monolithic (see Theorem 2.7). So:
Corollary 4.4. If Cp(X) is a monotonically monolithic space (resp., has
the Collins-Roscoe property) and Y is a cozero subset of X, then Cp(Y ) is
monotonically monolithic (resp., has the Collins-Roscoe property).
Corollary 4.5. If X is a monotonically stable space and Y is a cozero
subset of X, then Y is monotonically stable.
5. Monotone κ-monolithicity and function spaces
In this section we use the ideas in [38] to give a characterization of mono-
tonically κ-monolithic spaces and use it to prove that any monotonically
κ-monolithic space of tightness κ is monotonically monolithic. As a conse-
quence, we explain one result in [1] by proving that: If X is a monotonically
ω-monolithic countably compact space, then X is a monotonically monolithic
Corson compact space and has the Fréchet-Urysohn property. Finally, we find
some spaces X for which Cp(X) is monotonically κ-monolithic. These spaces
will be used in the next section to obtain spaces Cp(X) with the hereditary
D-property.
Proposition 5.1. A space X is monotonically κ-monolithic if and only if
one can assign to each finite subset F of X a countable collection N (F ) of
subsets of X such that, for every set A ⊂ X with |A| ≤ κ,
⋃
{N (F ) : F ∈
[A]<ω} contains an external network for cl(A).
48 II. MONOTONE MONOLITHICITY
Proof. If one can assign to each finite subset F of X a countable col-
lection N (F ) of subsets of X such that
⋃
{N (F ) : F ∈ [A]<ω} contains a
network for cl(A) for every subset A of X with |A| ≤ κ, then we denote
O(A) =
⋃
{N (F ) : F ∈ [A]<ω} for every subset A of X with |A| ≤ κ. It is
easy to see that operator O is a monotonically κ-monolithic operator for X,
as in Definition 9.1 in Chapter I.
LetX be a monotonically κ-monolithic space and letO be a monotonically
κ-monolithic operator for X as in Definition 9.1 in Chapter I. If F ⊂ X and
|F | < ω, then N (F ) = O(F ) is a countable family of subsets of X. We show
that
⋃
{N (F ) : F ∈ [A]<ω} contains a network for cl(A) for every subset
A of X with |A| ≤ κ. Let A ⊂ X with |A| ≤ κ. If x ∈ cl(A) and U is
an open neighborhood of x, then x ∈ N ⊂ U for some N ∈ O(A). Let
F ⊂ A have minimal cardinality such that N ∈ O(F ). We claim that F is
finite. Suppose otherwise, and let F = {xα : α < λ} where λ = |F |. Now
let Fα = {xβ : β < α}. Since F =
⋃
{Fα : α < λ} then, by condition 3)
in the Definition 9.1 of Chapter I, we have O(F ) =
⋃
{O(Fα) : α < λ}; so
N ∈ O(Fα) for some α < λ. But |Fα| < |F |; this is a contradiction. Thus,
F ∈ [A]<ω and N ∈ O(F ) = N (F ). 
Proposition 5.2. If X is monotonically κ-monolithic and t(X) ≤ κ, then
X is monotonically monolithic.
Proof. By Proposition 5.1, we can assign to each finite subset F ⊂ X, a
countable collection N (F ) such that, for each subset A of X with |A| ≤ κ, the
family
⋃
{N (F ) : F ∈ [A]<ω} is an external network for cl(A). By Theorem
7.7 in Chapter I, in order to get a proof of our proposition, it is enough to
prove that for each A ⊂ X the family
⋃
{N (F ) : F ∈ [A]<ω} is an external
network of cl(A). So, take A ⊂ X and x ∈ cl(A). Let U be an open subset
of X with x ∈ U . Since t(X) ≤ κ, there is B ⊂ A with |B| ≤ κ such that
x ∈ cl(B). Because
⋃
{N (F ) : F ∈ [B]<ω} is an external network of cl(B),
we can take a finite set F ⊂ B ⊂ A, and N ∈ N (F ) such that x ∈ N ⊂ U .
This proves that
⋃
{N (F ) : F ∈ [A]<ω} is an external network for cl(A). 
O. Alas, V.V. Tkachuk and R.G. Wilson asked in [1] if X is monolithic
whenever X is compact and monotonically ω-monolithic. The following result
shows that the answer is positive.
Corollary 5.3. If X is a Corson compact monotonically ω-monolithic
space, then X is monotonically monolithic.
Proof. Let X be a Corson compact space. Then X is contained in a
Σ-product of real lines, that is, X is contained in Cp(Lκ), where Lκ is the
one-point Lindelöfication of a discrete space of cardinality κ. It is easy to se
that Ln
κ is Lindelöf for each n ∈ ω. By the Theorem of Arhangel’skii-Pykteev
we have that t(Cp(Lκ)) ≤ ω and so t(X) ≤ ω. Since X is monotonically ω-
monolithic, Proposition 5.1 guarantees that X is monotonically monolithic.

5. MONOTONE κ-MONOLITHICITY AND FUNCTION SPACES 49
Since every compact monotonically ω-monolithic space is a Corson com-
pact space (see Corollary 11.3 in Chapter I), we obtain:
Corollary 5.4. If X is a compact monotonically ω-monolithic space, then
X is a monotonically monolithic Corson compact space.
Theorem 5.5. If X is a monotonically ω-monolithic countably compact
space, then X is a monotonically monolithic Corson compact space and has
the Fréchet-Urysohn property.
Proof. LetX be a monotonically ω-monolithic countably compact space.
By Proposition 9.11 in Chapter I, X is compact and has the Fréchet-Urysohn
property. By Corollary 5.4, X is a monotonically monolithic Corson compact
space. 
In the second part of this section we find some spaces for which Cp(X) is
monotonically κ-monolithic, but before we need to prove some results.
Proposition 5.6. Given an infinite cardinal κ, suppose that X is a mono-
tonically κ-monolithic space and f is a continuous function from X onto Y
such that f(cl(A)) = cl(f(A)) for any set A ⊂ X with |A| ≤ κ. Then Y is
monotonically κ-monolithic.
Proof. Fix a point xy ∈ f−1(y) for each y ∈ Y and let O be an oper-
ator which witnesses monotone κ-monolithicity of X, as in Definition 9.1 in
Chapter I. Given a set B ⊂ Y with |B| ≤ κ let BX = {xy : y ∈ B} and
N (B) = {f(N) : N ∈ O(BX)}. It is routine to see that N satisfies the con-
ditions (1)−(3) of the definition of monotone κ-monolithicity for the space
Y .
To see that N (B) is an external network of cl(B) in Y ; take any y ∈ cl(B)
and U an open set in Y with y ∈ U . The set BX being of cardinality at most
κ we have f(cl(BX)) = cl(B), so there is a point x ∈ cl(BX) with f(x) = y.
If N ∈ O(BX) is such that x ∈ N ⊂ f−1(U), then f(N) ∈ N (B) and
y ∈ f(N) ⊂ U . Thus, Y is monotonically κ-monolithic. 
Proposition 5.7. Let X be a space and let Y be a Gκ-dense subset of
X which is C-embedded in X. Then for every A ⊂ Cp(X) with |A| ≤ κ the
function πY ↾ A : A→ πY (A) is a homeomorphism.
Proof. The function π = πY ↾ A : A → πY (A) is bijective and con-
tinuous. We shall prove that π is open. Fix an open subset B of A. Let
f ∈ B. Take W = [x1, . . . , xn;U1 . . . , Un] a canonical open set in Cp(X) with
f ∈ (W ∩A) ⊂ B. Notice that Gi
f = f−1(f(xi)) and G
i
g = g−1(g(xi)) for each
g ∈ A\B, are sets of typeGδ inX. Thus, the setsGi =
⋂
{Gi
f∩G
i
g : g ∈ A\B}
are nonempty sets of type Gκ in X. Since Y is a Gκ-dense subset of X, we
can choose yi ∈ Y ∩ Gi for i = 1, . . . , n. Finally, V = [y1, . . . , yn;U1 . . . , Un]
is canonical open set in Cp(Y ) with π(f) ∈ (V ∩ π(A)) ⊂ π(B). 
50 II. MONOTONE MONOLITHICITY
Proposition 5.8. If Cp(X) is a monotonically κ-monolithic space and Y
is a Gκ-dense subset of X which is C-embedded in X, then Cp(Y ) is mono-
tonically κ-monolithic.
Proof. By Proposition 5.7, the function πY | A : A→ πY (A) is a home-
omorphism for every A ⊂ Cp(X) with |A| ≤ κ. It is easy to see in this case
that πY (cl(A)) = cl(πY (A)) for any set A ⊂ Cp(X) with |A| ≤ κ. It follows
from Proposition 5.6, that Cp(Y ) is monotonically κ-monolithic. 
Given a space X and an infinite cardinal κ, recall that the Hewitt-κ-
extension of X that we denote by υκX is the subspace of βX consisting of all
points x ∈ βX for which every set of type Gκ in βX containing x intersectsX.
It is well known and easy to prove that any continuous function f : X → R
can be extended continuously on υκX (see [29]).
Theorem 5.9. If X is a space such that its Hewitt κ-extension is mono-
tonically κ-stable, then X is monotonically κ-stable, that is, Cp(X) is mono-
tonically κ-monolithic.
Proof. Since υκX is monotonically κ-stable, it follows from Remark 2.10
for Theorem 2.7 that Cp(υκX) is monotonically κ-monolithic. Since X is
Gκ-dense and is C-embedded in υκX, we can apply Proposition 5.8 to see
that Cp(X) is monotonically κ-monolithic. It follows from Remark 2.10 for
Theorem 2.7 that X is monotonically κ-stable. 
Lemma 5.10. Let λ be an infinite cardinal number and κ = λ+. If X is
a product of Lindelöf Σ-spaces and Y is a Σκ-product in X then υλY = X.
Proof. Suppose X =
∏
{Xt : t ∈ T} where each Xα is a Lindelöf Σ-
space and Y = Σκ
∏
{Xt : t ∈ T}. Clearly Y contains σ
∏
{Xt : t ∈ T}.
We shall prove that Y is C-embedded in X. Indeed, if f : Y → R is a
continuous function. Since l(
∏
{Xα : α ∈ K}) ≤ ω for each finite set K ⊂ T
and iw(R) ≤ ω, by Lemma 3.8 there exists a countable set S ⊂ κ such that
if x, x′ ∈ Y and pS(x) = pS(x
′) then f(x) = f(x′). Since pS ↾ Y : Y → pS(Y )
is a quotient function, by Lemma 3.1 there exists g : πS(Y ) → R, continuous,
such that f = g ◦ pS ↾ Y . Thus f̃ = g ◦ pS : X → R is continuous and
f = f̃ ↾ Y .
Since Y is C-embedded in X, βY = βX. It is well known that any
Lindelöf space is realcompact and that an arbitrary product of realcompact
spaces is realcompact (see [29]). So X is realcompact, that is, X = υωX.
Therefore, for any x ∈ βX \ X there exists an open set U ⊂ βX such that
U ∩ Y ⊂ U ∩ X = ∅. This shows that υλY ⊂ X. Now we shall prove the
other contention. Take x ∈ X and let U be a set of type Gλ in βX such that
x ∈ U . Since U ∩X is a non-empty set of type Gλ in X and Y is Gλ-dense
in X, (U ∩ Y ) = (U ∩X) ∩ Y 6= ∅. Thus x ∈ υλY . Therefore X ⊂ υλY . 
It follows from Proposition 9.3 in Chapter I and Remark 2.10 for Theorem
2.7 that a space X is monotonically stable if and only if it is λ-stable for any
5. MONOTONE κ-MONOLITHICITY AND FUNCTION SPACES 51
cardinal λ. The following result follows directly from Corollary 3.11, Theorem
5.9 and Lemma 5.10.
Corollary 5.11. Let λ be an infinite cardinal number and κ = λ+. If X
is a product of Lindelöf Σ spaces and Y is a Σκ-product in X, then Cp(Y ) is
monotonically λ-monolithic.
Given a cardinal κ, recall that a topological space X is κ-pseudocompact
if it is Gκ-dense in βX. It is well known that a space X is κ-pseudocompact
if and only if for each continuous function f : X → R
κ the space f(X) is
compact.
Since any κ-pseudocompact space X is pseudocompact, then it is C-
embedded in βX. The following result is a consequence of Proposicion 5.8 or
Theorem 5.9.
Corollary 5.12. For every κ-pseudocompact space X, the space Cp(X) is
monotonically κ-stable.
Corollary 5.13. Let λ be an infinite cardinal number and κ = λ+, then
Cp([0, κ)) is monotonically λ-monolithic.
In [77], the following questions were proposed: (1) Suppose that Cp(X)
has the Collins-Roscoe property, must X be Lindelöf? (2) Suppose that
Cp(X) has the Collins-Roscoe property, must X be Lindelöf Σ? Now we are
going to use the results obtained in this section to find some counterexamples
for the first (hence for the second) above question.
Example 5.14. Cp([0, ω1)) is a Collins-Roscoe space (see Definition 10.1
in Chapter I). Indeed, for each α < ω1, let Cα
p ([0, ω1)) be the set of all
the real-valued continuous functions defined on [0, ω1) which are constant
on (α, ω1). It is a well known fact that Cp([0, ω1)) =
⋃
{Cα
p ([0, ω1)) : α <
ω1}. Moreover, for each α < ω1, C
α
p ([0, ω1)) is homeomorphic to Cp([0, α])×
R. Thus, nw(Cα
p ([0, ω1))) = nw(Cp([0, α])) = nw([0, α]) = ω. Therefore,
Cp([0, ω1)) is the union of ω1 subspaces with countable network. Since ω1 is
pseudocompact, by Proposition 5.12, the space Cp([0, ω1)) is monotonically
ω-monolithic. So, by Theorem 10.4 in Chapter I, Cp([0, ω1)) has the Collins-
Roscoe property.
Proposition 5.15. If X is a monotonically ω-monolithic space of cardi-
nality ω1, then Cp(Cp(X)) is a Collins-Roscoe space.
Proof. Let X = {xα : α < ω1}. For each α < ω1 let Xα = {xβ : β ≤ α}.
For each α < ω1, let Zα = π∗
Xα
(Cp(Xα)) ⊂ Cp(Cp(X)) the set of all continu-
ous functions in Cp(Cp(X)) which have a factorization trough of Cp(Xα). Let
f ∈ Cp(Cp(X)), by Lemma 1.5, there exist a countable set S ⊂ X and a con-
tinuous real-valued function g on πS(Cp(X)) such that f = g ◦ πS. Since S is
countable, there exists α < ω1 such that S ⊂ Xα. Let π
Xα
S : Cp(Xα) → Cp(S)
the restriction function. Then we have f = h ◦ πXα
= π∗
Xα
(h) where h =
52 II. MONOTONE MONOLITHICITY
g ◦ πXα
S ∈ Cp(Xα). Hence f ∈ Zα. Therefore Cp(Cp(X)) ⊂
⋃
{Zα : α < ω1}.
Moreover, for each α < ω1, Zα is homeomorphic to Cp(Xα) (see Proposi-
tion I.2.2). Since Xα is countable, nw(Zα) = nw(Cp(Xα)) = nw(Xα) = ω.
Therefore, Cp(Cp(X)) is the union of ω1 subspaces with countable network.
Moreover, by Remark 1.9 for Theorem 1.6, the space Cp(Cp(X)) is monoton-
ically ω-monolithic. So, by Theorem 10.4 in Chapter I, the space Cp(Cp(X))
has the Collins-Roscoe property. 
Example 5.16. Let Lω1
be the one-point Lindelöfication of a discrete
space of cardinality ω1. As we have seen in Example 1.10, Lω1
is monoton-
ically monolithic and Cp(Lω1
) is not Lindelöf. On the other hand, since the
cardinality of Lω1
is ω1, by Proposition 5.15, the space Cp(Cp(Lω1
)) has the
Collins-Roscoe property.
6. Monotone <κ-monolithicity
In this section, we are going to introduce the concept of monotonically
<κ-monolithic space when κ ≥ ω1. This concept is weaker than that of mono-
tonically κ-monolithic space. In the previous section we have proved that a
monotonically κ-monolithic space of tightness κ is monotonically monolithic.
Thus, we can ask if there is a relation between monotone κ-monolithicity (or
monotone <κ-monolithicity) and other cardinal functions. We know that any
monotonically monolithic space is hereditarily D. Our main result in this sec-
tion states that if X is a monotonically <κ-monolithic space and nw(X) ≤ κ,
then X is hereditarily D. Actually, this has been the purpose of introducing
the notion of monotonically <κ-monolithic space. We will use our main result
to obtain spaces Cp(X) with the hereditary D-property. In particular, we are
going to show that Cp(X) is hereditarily D whenever X is a pseudocompact
space with network weight at most ω1.
Definition 6.1. For a cardinal number κ > ω1, we say that a space X is
monotonically <κ-monolithic if for each finite subset F of X we can assign a
countable collection N (F ) of subsets of X such that, for each subset A ⊂ X
with |A| < κ, the family
⋃
{N (F ) : F ∈ [A]<ω} is an external network for
cl(A).
Let X be a monotonically <κ-monolithic space with operator N , and let
φ be a neighborhood assignment on X. We define the relation R = R(N , φ)
from X to [X]<ω as follows: (x, F ) ∈ R if and only if there is N ∈ N (F )
with x ∈ N ⊂ φ(x). For any A ⊂ X we will use the following notation
R(A) =
⋃
{R−1(F ) : F ∈ [A]<ω} = R−1([A]<ω). Clearly, R(A) ⊂ R(B)
whenever A ⊂ B ⊂ X.
Given a neighborhood assignment φ onX, recall that for A ⊂ X we denote
by φ(A) the set
⋃
{φ(x) : x ∈ A}, moreover, a subset Z of X is φ-close if
Z ⊂ φ(x) for each x ∈ Z.
6. MONOTONE <κ-MONOLITHICITY 53
Remarks 6.2. Let X be a monotonically <κ-monolithic space with op-
erator N and let φ be a neighborhood assignment on X, then:
(1) If A is a subset of X with |A| < κ and x ∈ cl(A), then there is
F ∈ [A]<ω such that (x, F ) ∈ R; that is, cl(A) ⊂ R(A).
(2) If F ∈ [X]<ω, then we have that R−1(F ) =
⋃
{ZN : N ∈ N (F )}
where ZN = {x ∈ X : x ∈ N ⊂ φ(x)}. Observe that ZN is φ-close
for each N ∈ N (F ). Therefore, R−1(F ) is the union of a countable
collection of φ-close subsets.
(3) If {Eβ : β < α} is a collection of subsets of X, then R(
⋃
{Eβ : β <
α}) =
⋃
{R(
⋃
{Eγ : γ ≤ β}) : β < α}.
(4) Let {Eβ : β < α} be a collection of subsets of X, let {Uβ : β < α}
be a family of open subsets of X. Assume that for each β < α we
have that R(
⋃
{Eγ : γ ≤ β}) ⊂
⋃
{Uγ : γ ≤ β}, then for each
β ≤ α we have R (
⋃
{Eγ : γ < β}) ⊂
⋃
{Uγ : γ < β}. Indeed, by the
previous remark, if β ≤ α we can see that: R (
⋃
{Eγ : γ < β}) =
⋃
{R(
⋃
{Eδ : δ ≤ γ}) : γ < β} ⊂
⋃
{
⋃
{Uδ : δ ≤ γ} : γ <
β} =
⋃
{Uγ : γ < β}. Moreover, by the first remark, if β ≤ α and
|
⋃
{Eγ : γ < β}| < κ, then cl(
⋃
{Eγ : γ < β}) ⊂
⋃
{Uγ : γ < β}.
In order to prove our main result we need the following lemmas.
Lemma 6.3. Let X be a topological space. If α is an ordinal number,
{Eβ : β < α} is a sequence of closed discrete subsets of X and {Uβ : β < α}
is a sequence of open sets in X such that Eβ ⊂ Uβ \
⋃
{Uγ : γ < β} for
each β < α, and cl(
⋃
{Eγ : γ < β}) ⊂
⋃
{Uγ : γ < β} for β ≤ α, then
⋃
{Eβ : β < α} is closed and discrete in X.
Proof. Let x in E = cl(
⋃
{Eβ : β < α}) ⊂
⋃
{Uβ : β < α}. Let β the
first ordinal number such that x ∈ Uβ. Then U = Uβ \ cl(
⋃
{Eγ : γ < β}) is
an open neighborhood of x such that U ∩ E ⊂ Eβ. Since Eβ is closed and
discrete, it follows that x ∈ Eβ. Then we can select an open neighborhood
V of x such that V ∩ Eβ = {x}. Thus W = U ∩ V is an open neighborhood
of x such that W ∩ E = {x}. This shows that
⋃
{Eβ : β < α} is closed and
discrete in X. 
Lemma 6.4. Let X be a monotonically <κ-monolithic space with operator
N , and let φ be a neighborhood assignment on X. If D ⊂ X is countable,
closed and discrete with D ⊂ V for an open subset V ⊂ X and if x∗ ∈ X \V ,
then there exists a countable closed and discrete subset E ⊂ X \ V of X with
x∗ ∈ E and such that R(D ∪ E) ⊂ V ∪ φ(E).
Proof. Take a family {Ωn : n ∈ ω} of infinite disjoint subsets of ω such
that ω =
⋃
{Ωn : n ∈ ω} and {0, . . . , n} ⊂ Ω1 ∪ . . . ∪ Ωn. We construct the
set E in a recursive process.
Step 0. Let E0 = {e0} where e0 = x∗ and let {Zi : i ∈ Ω0} be a countable
collection of φ-close subsets of X such that R(D ∪E0) ⊂
⋃
{Zi : i ∈ Ω0} (see
Remark 6.2 (2)).
54 II. MONOTONE MONOLITHICITY
Suppose that for every k ≤ n we have chosen countable closed and discrete
sets Ek and constructed families of φ-close subsets {Zi : i ∈ Ωk} such that
for Uk = V ∪ φ(Ek) we have:
H(k) : Ek ⊂ Uk \
⋃
{Ui : i < k};
I(k) : R(D ∪
⋃
{Ei : i ≤ k}) ⊂
⋃
{Zi : i ∈ Ωk};
J(k) :
⋃
{Zi : i < k} ⊂
⋃
{Ui : i ≤ k}.
Step n + 1. If Zi ⊂
⋃
{Uk : k ≤ n} for each i ∈
⋃
{Ωk : k ≤ n}, then
E =
⋃
{Ek : k ≤ n} satisfies the requested conditions and so we would finish
the proof. Otherwise, let m be the first natural number in
⋃
{Ωk : k ≤ n}
such that Zm \
⋃
{Uk : k ≤ n} 6= ∅. Notice that by J(n), m ≥ n. Fix
en+1 ∈ Zm \
⋃
{Uk : k ≤ n}. Take En+1 = en+1. Because of Remark 6.2 (2),
we can take a collection {Zi : i ∈ Ωn+1} of φ-close subsets of X such that
R(D ∪
⋃
{Ei : i ≤ n + 1}) ⊂
⋃
{Zi : i ∈ Ωn+1}. Observe that conditions
H(n+1) and I(n+1) hold. We prove that J(n+1) holds. Since J(n) holds, it
is enough to prove that Zn ⊂
⋃
{Uk : k ≤ n+1}. If n < m, since n ∈
⋃
{Ωk :
k ≤ n} by the choice of m, we have Zn ⊂
⋃
{Uk : k ≤ n}. If n = m, since Zn
is φ-close and by the choice of En+1, we have Zn ⊂ φ(En+1) ⊂ V ∪ φ(En+1).
Therefore J(n+ 1) holds.
Assume that we cannot finish the recursive process in a finite step, then
we have, for each n ∈ ω, a non-empty set En and a family {Zi : i ∈ Ωn} of
φ-close sets such that H(n), I(n) and J(n) hold. Take E =
⋃
{En : n ∈ ω}.
Notice that x∗ = e0 ∈ E. Since H(n) holds for every n ∈ ω, we have
E ⊂ X \ V . Because of I(n), J(n) and Remark 6.2 (3) we can see that
R(D ∪ E) =
⋃
{R(D ∪
⋃
{Ek : k ≤ n}) : n ∈ ω} ⊂
⋃
{Zn : n ∈ ω} ⊂
⋃
{Un :
n ∈ ω} = V ∪ φ(E). We shall prove that E is closed and discrete. Because
of Remark 6.2 (1), we have cl(E) ⊂ R(E) ⊂ R(D ∪ E) ⊂
⋃
{Un : n ∈ ω}.
Using this fact and H(n) for n ∈ ω, we conclude that cl(
⋃
{En : n < β}) ⊂
⋃
{Un : n < β} for β ≤ ω. Therefore, by Lemma 6.3 we conclude that the set
E =
⋃
{En : n < ω} is closed and discrete. 
Lemma 6.5. Let X be a monotonically <κ-monolithic space with operator
N and let φ be a neighborhood assignment on X. If D ⊂ X is closed and
discrete, |D| < κ, D ⊂ V for an open set V ⊂ X and x∗ ∈ X \ V , then there
exists E ⊂ X \ V , closed and discrete in X with |E| ≤ |D|, x∗ ∈ E and such
that R(D ∪ E) ⊂ V ∪ φ(E).
Proof. We proceed by transfinite induction on the cardinality of D. If
|D| = ω, the result follows from Lemma 6.4. Suppose that there is a cardinal
number λ < κ such that the result is true for every subset D with cardinality
less than λ. We are going to prove that the Lemma holds when we consider
subsets D of cardinality λ. So, take a closed and discrete subset D = {xα :
α < λ} of X such that D ⊂ V for some open subset V ⊂ X and x∗ ∈ X \ V .
We construct the set E by transfinite recursion.
6. MONOTONE <κ-MONOLITHICITY 55
Step 0. Let D0 = {x0}, and take x∗0 = x∗. By Lemma 6.4, there is a
countable closed and discrete subset E0 of X with E0 ⊂ X \ V , x∗0 ∈ E0 and
such that R({x0} ∪ E0) ⊂ V ∪ φ(E0).
Assume that α < λ and for each β < α, we have constructed closed and
discrete subsets Eβ such that for Uβ = V ∪ φ(Eβ), the following conditions
hold:
H(β) : |Eβ| ≤ max{ω, |β|};
I(β) : Eβ ⊂ Uβ \
⋃
{Uγ : γ < β};
J(β) : R({xγ : γ ≤ β} ∪
⋃
{Eγ : γ ≤ β}) ⊂
⋃
{Uγ : γ ≤ β}.
Step α. Since H(β) and J(β) hold for each β < α, by Remark 6.2 (4) we
have cl(
⋃
{Eγ : γ < β}) ⊂
⋃
{Uγ : γ < β} for each β ≤ α. Using this fact and
I(β) for each β < α, by Lemma 6.3, we conclude that
⋃
{Eβ : β < α} is closed
and discrete in X. Then Dα = {xβ : β ≤ α} ∪
⋃
{Eβ : β < α} is closed and
discrete in X. Besides, |Dα| ≤ max{ω, |α|}. If V ∪ φ(
⋃
{Eβ : β < α}) = X,
then the set E =
⋃
{Eβ : β < α} satisfies the required conditions and the
proof would be finished. In the other possible case, fix x∗α ∈ X\
⋃
{V ∪φ(Eβ) :
β < α}. By hypothesis, there exists Eα ⊂ X \
⋃
{V ∪ φ(Eβ) : β < α}, closed
and discrete in X with |Eα| ≤ |Dα| ≤ max{ω, |α|}, x∗α ∈ Eα and such that
R(Dα ∪ Eα) ⊂
⋃
{V ∪ φ(Eβ) : β ≤ α}. Note that, for Uα = V ∪ φ(Eα),
conditions H(α), I(α) and J(α) hold.
Finally, if we do not finish the process in a step α < λ, we take E =
⋃
{Eα :
α < λ}. Clearly, E ⊂ X\V , |E| ≤ λ = |D| and x∗ ∈ E. Moreover, using J(β)
for β < λ and the first part of Remark 6.2 (3) we haveR(D∪E) =
⋃
{R({xγ :
γ ≤ β}∪
⋃
{Eγ : γ ≤ β}) : β < λ} ⊂
⋃
{Uβ : β < λ} = V ∪φ(E). So, we only
need to prove that E is closed and discrete. Since H(β) and J(β) hold for
each β < λ, by Remark 6.2 (4) we have cl(
⋃
{Eγ : γ < β}) ⊂
⋃
{Uγ : γ < β}
for each β ≤ λ. Using this fact and I(β) for each β < λ, by Lemma 6.3, we
conclude that E =
⋃
{Eβ : β < λ} is closed and discrete in X. 
Every monotonically monolithic space is monotonically κ-monolithic for
any infinite cardinal κ (see Proposition 9.3 in Chapter I). It is clear from
Definition 6.1 and Proposition 5.1 in Chapter I that every monotonically κ-
monolithic space is monotonically <κ-monolithic for all κ ≥ ω1. So, the
following result is a generalization of Theorem 7.6 in Chapter I.
Theorem 6.6. Let X be a monotonically <κ-monolithic space where κ ≥
nw(X). Then, X is hereditarily D.
Proof. It is sufficient to prove that X has property D. Let φ be a
neighborhood assignment in X, and let N = {Nα : α < κ} be a network for
X. We will construct by transfinite recursion a closed and discrete subset E
of X such that φ(E) = X as follows:
Step 0. Let D0 = V = ∅ and ξ0 be the first ordinal ξ < κ such that there
exists x ∈ X with x ∈ Nξ0 ⊂ φ(x). Fix x∗0 ∈ X such that x∗0 ∈ Nξ0 ⊂ φ(x∗0).
Because of Lemma 6.4, there is a countable closed and discrete set E0 ⊂ X
with x∗0 ∈ E0 and such that R(E0) ⊂ φ(E0).
56 II. MONOTONE MONOLITHICITY
Let α < κ and assume that for each β < α we have chosen a point x∗β ∈ X,
an ordinal ξβ < κ and a closed discrete subset Eβ ⊂ X; in such form that for
Uβ = φ(Eβ) the following conditions are satisfied:
H(β) : ξγ < ξβ whenever γ < β < α;
I(β) : |Eβ| ≤ max{ω, |β|};
J(β) : Eβ ⊂ Uβ \
⋃
{Uγ : γ < β};
K(β) : R(
⋃
{Eγ : γ ≤ β}) ⊂
⋃
{Uγ : γ ≤ β};
L(β) : ξβ is the first element of κ such that there is x ∈ X \
⋃
{Uγ : γ < β}
with x ∈ Nξβ ⊂ φ(x). Also, x∗β is a fixed point in X \
⋃
{Uγ : γ < β} with
x∗β ∈ Eβ and x∗β ∈ Nξβ ⊂ φ(x∗β).
Step α. Since I(β) and K(β) hold for each β < α, by Remark 6.2 (4)
we have cl(
⋃
{Eγ : γ < β}) ⊂
⋃
{Uγ : γ < β} for each β ≤ α. Using this
fact and J(β) for β < α, by Lemma 6.3, the set Dα =
⋃
{Eβ : β < α} is
closed and discrete in X. If φ(Dα) = X, then the set E = Dα satisfies the
required conditions and we would finish the recursive process. Otherwise,
observe that |Dα| ≤ max{ω, |α|}. Let ξα be the first ordinal ξ such that there
is x ∈ X \ φ(Dα) with x ∈ Nξ ⊂ φ(x). Choose x∗α ∈ X \ φ(Dα) such that
x∗α ∈ Nξα ⊂ φ(x∗α). By Lemma 6.5, there exists a closed and discrete set
Eα ⊂ X \ φ(Dα) which satisfies |Eα| ≤ |Dα| ≤ max{ω, |α|}, x∗α ∈ Eα and
such that R(Dα∪Eα) ⊂ φ(Dα∪Eα). Note that conditions H(α), I(α), J(α),
K(α) and L(α) hold.
Finally, if we do not finish the process in a step α < κ, take E =
⋃
{Eα :
α < κ}. If x ∈ X \ φ(E) = X \ {Uγ : γ < κ}, then there exists ξ < κ with
x ∈ Nξ ⊂ φ(x) and there is β < κ with ξβ > ξ; this contradicts the choice of
ξβ. Therefore, we must have that φ(E) = X. So cl(
⋃
{Eγ : γ < β}) ⊂
⋃
{Uγ :
γ < β} for each β ≤ κ. Using this fact and J(β) for β < κ, by Lemma 6.3,
the set E is closed and discrete in X. 
We can also to define monotone <κ-stability as follows:
Definition 6.7. For a cardinal number κ > ω1, we say that a topological
space X is monotonically <κ-stable if and only if for each finite collection
F ⊂ Cp(X) we can assign a countable collection N (F ) of subsets of X such
that for every A ⊂ Cp(X) with |A| < κ the family
⋃
{N (F ) : F ∈ [A]<ω} is
a network for ∆cl(A)(X) modulo ∆cl(A).
Remark 6.8. As in Remark 1.9 we can see that statements in Theo-
rem 1.6, Corollary 1.7 and Proposition 1.8 remain true if we replace mono-
tone monolithicity by monotone <κ-monolithicity. Also, as in Remark 2.10,
statements in Theorem 1.6, Theorem 2.7, Corollary 2.8 and Corollary 2.9 re-
main true if we replace monotonically monolithic and monotonically stable
by monotonically <κ-monolithic and monotonically <κ-stable, respectively.
As a consequence of Theorem 6.6 and the previous remark we have:
Corollary 6.9. Let X be a monotonically <κ-stable space where κ ≥
nw(X). Then, Cp(X) is hereditarily D.
6. MONOTONE <κ-MONOLITHICITY 57
Corollary 6.10. Let X be a monotonically <κ-monolithic space where
κ ≥ nw(X). Then, Cp,2n+1(X) is hereditarily D for any n ∈ ω.
Corollary 6.11. Let X be a monotonically <κ-monolithic space where
κ ≥ nw(X) and n ∈ ω. Then, every closed continuous image Y of a subspace
of Cp,2n(X) satisfies l(Y ) = e(Y ).
Corollary 6.12. Let X be a monotonically <κ-monolithic space where
κ ≥ nw(X), n ∈ ω and let Y be a closed continuous image of a subspace of
Cp,2n(X). Then Y is is compact whenever it is countably compact.
Given an infinite cardinal λ, recall that a space X is initially λ-compact if
every open cover U of X with |U| ≤ λ has a finite subcover. Given a cardinal
κ, a topological space X is called initially <κ-compact if X is initially λ-
compact for every λ < κ.
Theorem 6.13. If X has a cover K and a countable network N modulo K
such that each K ∈ K is initially <κ-compact, then Cp(X) is monotonically
<κ-monolithic.
Proof. For every family E of subsets of X, we take C(E) = {N \E : E ∈
E , N ∈ N} and F(E) = {
⋃
E0 : E0 ⊂ E , |E0| < ω}. For each finite set F ⊂
Cp(X), take EF = {g−1(B) : g ∈ F,B ∈ B(R)} and N (F ) = W(C(F(EF )))
(see Remark 1.1). Observe that, since F is finite, N (F ) is countable. We
are going to prove that for each A ⊂ Cp(X) satisfying |A| < κ, the family
⋃
{N (F ) : F ∈ [A]<ω} is an external network of cl(A) in Cp(X). Take
A ⊂ Cp(X) with |A| < κ and f ∈ cl(A).
Claim. If x ∈ X, B ∈ B(R) and f(x) ∈ B, then there are a finite set
F ⊂ A and P ∈ C(F(EF )) such that x ∈ P and f ∈ [P ;B].
We will prove the Claim. Let K ∈ K which contains x. For each y ∈ K \
f−1(B), we take a set By ∈ B(R) such that f(y) ∈ By and f(x) 6∈ cl(By), and
we take a function gy ∈ A such that gy(x) ∈ B \ cl(By) and gy(y) ∈ By. The
family {g−1
y (By) : y ∈ K \ f−1(B)} ∪ {f−1(B)} covers K and has cardinality
less than κ. By hypothesis, there is a finite set K0 ⊂ K \ f−1(B) such that
{g−1
y (By) : y ∈ K0} ∪ {f−1(B)} covers K. Take N ∈ N with K ⊂ N ⊂
(
⋃
{g−1
y (By) : y ∈ K0}) ∪ f
−1(B). By construction, if P = N \
⋃
{g−1
y (By) :
y ∈ K0}, then x ∈ P , P ∈ C(F(EF )) where F = {gy : y ∈ K0} ⊂ A and
f ∈ [P ;B]. This proves the Claim.
Now, assume that f ∈ U for an open subset U of Cp(X). Take x1, . . . , xn ∈
X and B1, . . . , Bn ∈ B(R) such that f ∈ [x1, . . . , xn;B1, . . . , Bn] ⊂ U . Be-
cause of the Claim, for each i = 1, . . . , n, there exist Fi ⊂ A, finite, and
Pi ∈ C(F(EFi
)) such that xi ∈ Pi and f ∈ [Pi;Bi]. Take F =
⋃
{Fi :
i = 1, . . . , n} ⊂ A. So, Pi ∈ C(F(EF )) for each i = 1, . . . , n. Finally, if
M = [P1, . . . , Pn;B1, . . . , Bn], then f ∈ M ⊂ [x1, . . . , xn;B1, . . . , Bn] ⊂ U ,
where M ∈ W(C(F(EF ))) = N (F ). 
Corollary 6.14. Let X be a countably compact and initially <nw(X)-
compact space. Then Cp(X) is hereditarily a D-space.
58 II. MONOTONE MONOLITHICITY
Given a cardinal λ, recall that a topological space X is λ-bounded if each
A ⊂ X which satisfies |A| ≤ λ is contained in a compact subset of X. Every
λ-bounded space is initially λ-compact [68].
Example 6.15. Let κ be a regular cardinal. Clearly, nw([0, κ)) = κ. Let
A ⊂ κ with |A| < κ. Then, A is contained in the compact subset [0, sup(A)] of
[0, κ). This shows that [0, κ) is λ-bounded and hence initially λ-compact, for
each cardinal λ < κ. So, [0, κ) is initially <κ-compact. Therefore, Cp([0, κ))
is monotonically <κ-monolithic (see Theorem 6.13). Since nw(Cp([0, κ))) =
nw([0, κ)) = κ, by Theorem 6.6, Cp([0, κ)) is hereditarily a D-space.
It follows from Definition 6.1 and Proposition I.5.1 that every monotoni-
cally κ-monolithic space is monotonically <κ+-monolithic. Then:
Corollary 6.16. If X is monotonically κ-monolithic and nw(X) ≤ κ+,
then X is hereditarily D.
In [16], R. Buzyakova asked if for every countably compact space X,
Cp(X) is hereditarily D. Afterward, she answered this question in the nega-
tive [19] by showing that there is a countably compact space X of cardinality
ω2 for which l(Cp(X)) > e(Cp(X)), and then Cp(X) is not a D-space. Nev-
ertheless, we have:
Proposition 6.17. If X is a pseudocompact space with network weight at
most ω1, then Cp(X) is hereditarily D.
Proof. Let X be a pseudocompact space with nw(X) ≤ ω1. By Corol-
lary 5.12, we have that Cp(X) is monotonically ω-monolithic. Because of
Corollary 6.16 and since nw(Cp(X)) = nw(X) ≤ ω1, we conclude that Cp(X)
is hereditarily a D-space. 
Recall that a collection A of subsets of the natural numbers ω is an almost
disjoint family if each A in A is infinite, and for two different elements A,B ∈
A, |A ∩ B| < ω. A maximal almost disjoint family (mad family) is a maximal
element in the family of all the almost disjoint families with the containment
order.
A topological space X is a Mrówka space if it has the form ω ∪ A, where
A is an almost disjoint family, and its topology is generated by the following
base: each {n} is open for every n ∈ ω, and an open canonical neighborhood
of A ∈ A is of the form {A} ∪ B where B ⊂ ω and A \ B is finite. In this
case, we denote X by Ψ(A). For every almost disjoint family A, Ψ(A) is
a 0-dimensional locally compact first countable space, A is a closed discrete
subspace of Ψ(A) and A is dense. Moreover, Ψ(A) is pseudocompact if and
only if A is maximal (see [32]). So, Ψ(A) is not normal if A is an infinite
mad family.
Example 6.18. For a maximal almost disjoint family A, the Mrówka
space Ψ(A) is pseudocompact and not countably compact. Under CH,
7. MONOTONE MONOLITHICITY AND Σs-PRODUCTS 59
nw(Cp(Ψ(A))) = nw(Ψ(A)) ≤ ω1. Hence, by Corollary 6.17, Cp(Ψ(A)) is
hereditarily D.
7. Monotone monolithicity and Σs-products
As we saw in Chapter I: Every Collins-Roscoe space is monotonically
monolithic. It is clear that if X is a Σ-product of an uncountable family
of monotonically monolithic spaces with more than one point, then X is
not a monotonically monolithic space, since it contains a closed subspace
homeomorphic to the Σ-product of the product of ω1 copies of the discrete
space of cardinality two. On the other hand, every σ-product of a family of
monotonically monolithic spaces is monotonically monolithic, and every σ-
product of a family of Collins-Roscoe spaces has the Collins-Roscoe property.
The concept of Σs-product was introduced by G.A. Sokolov who proved that
a compact space X is a Gul’ko compact space if and only if X embeds into
a Σs-product of real lines. It is easy to see that every Σs-product of a family
{Xt : t ∈ T} of spaces based on a point a ∈
∏
{Xt : t ∈ T} contains the
respective σ-product based on a, and is contained in the respective Σ-product
based on a. G. Gruenhage proved that every Gul’ko compact space has the
Collins-Roscoe property and V.V. Tkachuk gave a generalization of this result
showing that every Σs-product of a family of second countable spaces has the
Collins-Roscoe property.
Taking into account the above results, in [75] the following questions were
posed: Is it true that every Σs-product of monotonically monolithic spaces is
monotonically monolithic? Is it true that every Σs-product of Collins-Roscoe
spaces has the Collins-Roscoe property? In this section, we answer these two
questions in the affirmative.
Let {Xt : t ∈ T} be a family of spaces and suppose that s = {Tn : n ∈ ω}
is a sequence of subsets of T . Let X =
∏
{Xt : t ∈ T} and fix a point a ∈ X.
Given any x ∈ X, A ⊂ X and E ⊂ T , let supp(x) = {t ∈ T : x(t) 6= a(t)},
supp(x,E) = supp(x) ∩ E and supp(A,E) =
⋃
{supp(x,E) : x ∈ A}. For
each x ∈ X, let Ωx be the set {n ∈ ω : |supp(x, Tn)| < ω}. Then the subspace
Z = {x ∈ X : T =
⋃
{Tn : n ∈ Ωx}} of X is called the Σs-product centered
in a with respect to the sequence s.
Remark 7.1. Let X =
∏
{Xt : t ∈ T} be a topological product, let a be
one fixed point in X and s = {Tn : n ∈ ω} a sequence of subsets of T .
(1) If x is an element of the Σs-product centered in a with respect to the
sequence s, then |supp(x)| ≤ ω; in fact, supp(x) =
⋃
{supp(x, Tn) :
n ∈ Ωx}.
(2) If s∗ is a sequence of subsets of T with s ⊂ s∗, then the Σs-product
centered in a with respect to the sequence s is contained in the Σs∗-
product centered in a with respect to the sequence s∗.
60 II. MONOTONE MONOLITHICITY
If s is a sequence of subsets of a set T , we define a relation R on T as
follows: we say that t1R t2 if for every E ∈ s we have that t1 ∈ E if and only
if t2 ∈ E.
Lemma 7.2. Let s be a sequence of subsets of T closed under complements
and finite intersections. If H1, . . . , Hn ∈ [T ]<ω are nonempty sets such that
for ti ∈ Hi and tj ∈ Hj we have tiR tj if and only if i = j, then we can find
a disjoint family {E1, . . . , En} ⊂ s such that Hi ⊂ Ei for i = 1, . . . , n.
Proof. If n = 2, for t1 ∈ H1 and t2 ∈ H2 we can find E ∈ s such that
t1 ∈ E and t2 ∈ T \E. Let E1 = E and E2 = T \E. Then, {E1, E2} satisfies
the required conditions. For n > 2 take H1, . . . , Hn as in the Lemma. For
every i, j ≤ n with i 6= j, take a disjoint family {Eij, E
′
ij} ⊂ s such that
Hi ⊂ Eij and Hj ⊂ E ′
ij. Now take Ei =
⋂
{Eij ∩ E
′
ji : j ≤ n and j 6= i} for
i = 1, . . . , n. Then the elements in the family {E1, . . . , En} ⊂ s are pairwise
disjoint and Hi ⊂ Ei for i = 1, . . . , n. 
Theorem 7.3. Every Σs-product of monotonically monolithic spaces is
monotonically monolithic.
Proof. Suppose that Xt is monotonically monolithic and fix the respec-
tive operator Nt for every t ∈ T . Suppose that s = {Tn : n ∈ ω} is a
sequence of subsets of T . Let X =
∏
{Xt : t ∈ T} and fix a point a ∈ X.
We must prove that the Σs-product Z centered in a with respect to the
sequence s is monotonically monolithic. Notice that since monotone mono-
lithicity is a hereditary property, by the Remark 7.1 (2), we can suppose
that the family s is closed under complements and finite intersections. Let
E(s) = {{E1, . . . , En} ∈ [s]<ω : Ei ∩ Ej = ∅ for i 6= j} be the family of all
finite subfamilies of s such that its elements are pairwise disjoint.
Now we are ready to construct a monotonic monolithicity operator in Z.
Fix a set A ⊂ Z. Take E ⊂ T , then by the Remark 7.1 (1) the set supp(A,E)
has cardinality at most λ, where λ = max{|A| , ω}. Let NE(A) be the family
of all sets of the form
∏
{Nt : t ∈ E}, where Nt ∈ Nt(pt(A)) for t ∈ F ,
Nt = {a(t)} for t ∈ E \F and F is a finite subset of supp(A,E). Notice that
the cardinality of NE(A) does not exceed λ. Finally, let
N (A) =
{
Z ∩
⋂
E∈F
p−1
E (NE) : NE ∈ NE(A) for every E ∈ F and F ∈ E(s)
}
.
Since E(s) is countable, the cardinality of N (A) is also at most λ. We will
show that the operator N has the required properties.
Claim 1. If A ⊂ B ⊂ Z then N (A) ⊂ N (B).
Let E ⊂ T . Clearly supp(A,E) ⊂ supp(B,E). By the election of Nt
and since pt(A) ⊂ pt(B) we have Nt(pt(A)) ⊂ Nt(pt(B)) for every t ∈ T .
Therefore, NE(A) ⊂ NE(B). Now take N = Z ∩
⋂
{p−1
E (NE) : E ∈ F} ∈
N (A) where NE ∈ NE(A) for every E ∈ F and F ∈ E(s). Since NE ∈
7. MONOTONE MONOLITHICITY AND Σs-PRODUCTS 61
NE(A) ⊂ NE(B) for every E ∈ F , we have that N ∈ N (B). Hence, N (A) ⊂
N (B).
Claim 2. If A =
⋃
{Aα : α < γ} ⊂ Z where Aβ ⊂ Aγ for every
α < β < γ, then N (A) =
⋃
{N (Aα) : α < γ}.
By Claim 1, it suffices to show that N (A) ⊂
⋃
{N (Aα) : α < γ}. First
we will show that if E ⊂ T and NE ∈ NE(A) then NE ∈ NE(AαE
) for some
αE < γ. In fact, let E ⊂ T and takeNE ∈ NE(A). ThenNE =
∏
{Nt : t ∈ E}
where Nt ∈ Nt(pt(A)) for t ∈ F , Nt = {a(t)} for t ∈ E \ F and F is a finite
subset of supp(A,E). Since supp(A,E) =
⋃
{supp(Aα, E) : α < γ} and
Nt(pt(A)) = Nt(
⋃
{pt(Aα) : α < γ}) =
⋃
{Nt(pt(Aα)) : α < γ} for every
t ∈ F , then we can choose αE < γ such that Nt ∈ Nt(pt(AαE
)) for every
t ∈ F and F is a finite subset of supp(AαE
, E). Thus, NE ∈ NE(AαE
).
Now take N = Z ∩
⋂
{p−1
E (NE) : E ∈ F} ∈ N (A) where NE ∈ NE(A) for
every E ∈ F and F ∈ E(s). For every E ∈ F we can take αE < γ with
NE ∈ NE(AαE
). Let α = max{αE : E ∈ F}. Then NE ∈ NE(Aα) for each
E ∈ F and therefore N ∈ N (Aα). So, N (A) ⊂
⋃
{N (Aα) : α < γ}.
Claim 3. For every A ⊂ Z the family N (A) is an external network for
clZ(A).
Fix A ⊂ Z, x ∈ clZ(A) and U an open set in Z with x ∈ U . Let
K ⊂ T a finite set and let {Wt : t ∈ K} be a family such that: Wt is open
in Xt for every t ∈ K, a(t) 6∈ Wt if x(t) 6= a(t) and x ∈ W ⊂ U where
W = {y ∈ Z : y(t) ∈ Wt for each t ∈ K}. Let {H1, . . . , Hn} be a partition of
K such that for ti ∈ Hi and tj ∈ Hj, we have that tiR tj if and only if i = j.
By Lemma 7.2 we can obtain a disjoint family {E ′
1, . . . , E
′
n} ∈ E(s) such that
Hi ⊂ E ′
i for i = 1, . . . , n.
Take i ∈ {1, . . . , n}. Let ti ∈ Hi. Since x ∈ Z, then T =
⋃
{Tm : m ∈ Ωx}
and so |supp(x, Tmi
)| < ω for some Tmi
∈ s with ti ∈ Tmi
. For Ei = E ′
i ∩ Tmi
,
we have that Fi = supp(x,Ei) is a finite set and Hi ⊂ Ei. For every t ∈ Fi\Hi
let Wt = Xt \ {a(t)}. Notice that x(t) ∈ Wt \ {a(t)} for every t ∈ Fi. For
an arbitrary t ∈ Fi, since x(t) ∈ clXt
(pt(A)) and Nt(pt(A)) is a network for
clXt
(pt(A)), we can choose Nt ∈ Nt(pt(A)) such that x(t) ∈ Nt ⊂ Wt. If
t ∈ Ei \ Fi let Nt = {a(t)}. Thus, since Fi is a finite subset of supp(A,Ei),
we have NEi
=
∏
{Nt : t ∈ Ei} ∈ NEi
(A).
Finally, F = {E1, . . . , En} ∈ E(s). Let N = Z ∩
⋂
{p−1
E (NE) : E ∈ F}.
Since NE ∈ NE(A) for every E ∈ F , then N ∈ N (A). For G =
⋃
{Fi ∪Hi :
i = 1, . . . , n} and V = {y ∈ Z : y(t) ∈ Wt for each t ∈ G} we have x ∈ N ⊂
V ⊂ W ⊂ U . 
Corollary 7.4. [1] Any σ-product of monotonically monolithic spaces is
monotonically monolithic.
Corollary 7.5. [77] Any countable product of monotonically monolithic
spaces is monotonically monolithic.
Corollary 7.6. If X is a Σs-product of cosmic spaces, then Cp,n(X) is
monotonically monolithic for any n ∈ ω.
62 II. MONOTONE MONOLITHICITY
Proof. Since every space with countable network is monotonically mono-
lithic, we have that X is monotonically monolithic. Because of Corollary 3.4,
the space Cp(X) is monotonically monolithic. By Corollary 1.8, it happens
that Cp,n(X) is monotonically monolithic for any n ∈ ω. 
We will finish this section by proving that every Σs-product of a family
of Collins-Roscoe spaces shares this property.
Recall that a space X has the Collins-Roscoe property if and only if for
each x ∈ X, a countable collection G(x) of subsets of X is assigned such that,
whenever x ∈ U and U open, there is an open set V with x ∈ V ⊂ U such
that, whenever y ∈ V , then x ∈ N ⊂ U for some N ∈ G(y).
Theorem 7.7. Every Σs-product of Collins-Roscoe spaces has the Collins-
Roscoe property.
Proof. Suppose that, for each t ∈ T , Xt is Collins-Roscoe and fix the
respective operator Gt for every t ∈ T . Suppose that s = {Tn : n ∈ ω} is a
sequence of subsets of T . Let X =
∏
{Xt : t ∈ T} and fix a point a ∈ X. We
must prove that the Σs-product Z centered in a with respect to the sequence
s has the Collins-Roscoe property. Since the Collins-Roscoe property is a
hereditary property then, by Remark 7.1 (2), we can suppose that the family
{Tn : n ∈ ω} is closed under complements and finite intersections. As before,
we associate sequence s with the family E(s) = {{E1, . . . , En} ∈ [s]<ω :
Ei ∩ Ej = ∅ for i 6= j}.
Now we are ready to construct an operator that witnesses the Collins-
Roscoe property in Z. Fix a point x ∈ Z. Let E ⊂ T . By the Remark 7.1
(1), the set supp(x,E) is countable. Let GE(x) be the family of all sets of the
form
∏
{Nt : t ∈ E} where Nt ∈ Gt(x(t)) for t ∈ F , Nt = {a(t)} for t ∈ E \F
and F is a finite subset of supp(x,E). Since each Gt(x(t)) is countable, the
family GE(x) is also countable. Finally, let
G(x) =
{
Z ∩
⋂
E∈F
p−1
E (NE) : NE ∈ GE(x) for every E ∈ F and F ∈ E(s)
}
.
Since E(s) is countable, the family G(x) is countable. We will show that the
operator G has the required properties.
Fix x ∈ Z and an open set U in Z with x ∈ U . We shall prove that there
is an open set V with x ∈ V ⊂ U such that, whenever y ∈ V , then x ∈ N ⊂ U
for some N ∈ G(y). LetK ⊂ T be a finite set and let {Wt : t ∈ K} be a family
such that: Wt is an open set in Xt for each t ∈ K, a(t) 6∈ Wt if x(t) 6= a(t),
and x ∈ W ⊂ U , where W = {y ∈ Z : y(t) ∈ Wt for each t ∈ K}. Let
{H1, . . . , Hn} be a partition of K such that for ti ∈ Hi and tj ∈ Hj we have
that tiR tj if and only if i = j. By Lemma 7.2 we can obtain a disjoint family
{E ′
1, . . . , E
′
n} ∈ E(s) such that Hi ⊂ E ′
i for i = 1, . . . , n. Take i = {1, . . . , n}.
Let ti ∈ Hi. Since x ∈ Z, then T =
⋃
{Tn : n ∈ Ωx} and so |supp(x, Tmi
)| < ω
for some Tmi
∈ s with ti ∈ Tmi
. Then for Ei = E ′
i∩Tmi
we have that Hi ⊂ Ei
7. MONOTONE MONOLITHICITY AND Σs-PRODUCTS 63
and Gi = Hi ∪ supp(x,Ei) ⊂ Ei is finite. For each t ∈ supp(x,Ei) \ Hi let
Wt = Xt \ {a(t)}. Clearly x(t) ∈ Wt \ {a(t)} for every t ∈ supp(x,Ei).
Observe that F = {E1, . . . , En} ∈ E(s). Let G =
⋃
{Gi : i = 1, . . . , n}.
Let t be an arbitrary element of G. By the election of Gt, and because x(t)
belongs to Wt, there is an open set Vt in Xt with x(t) ∈ Vt ⊂ Wt such that
whenever yt ∈ Vt, then x(t) ∈ Nt ⊂ Wt for some Nt ∈ Gt(yt). Let V =
{y ∈ Z : y(t) ∈ Vt for each t ∈ G}. Clearly x ∈ V ⊂ W . Fix y ∈ V . Take
i = {1, . . . , n}. Let Fi = Gi∩ supp(y, Ei) ⊂ Ei. Notice that supp(x,Ei) ⊂ Fi.
For an arbitrary t ∈ Fi, since y(t) ∈ Vt, we can choose Nt ∈ Gt(y(t)) such
that x(t) ∈ Nt ⊂ Wt. For t ∈ Ei \ Fi, let Nt = {a(t)}. Thus, since Fi is a
finite subset of supp(y, Ei) we have NEi
=
∏
{Nt : t ∈ Ei} ∈ GEi
(x).
Finally notice that if N = Z ∩
⋂
{p−1
E (NE) : E ∈ F}, then N ∈ G(y).
Furthermore, for W ′ = {y ∈ Z : y(t) ∈ Wt for each t ∈ G}, we have that x ∈
N ⊂ W ′ ⊂ W ⊂ U . 
Corollary 7.8. [78] Every σ-product of Collins-Roscoe spaces has the
Collins-Roscoe property.
Corollary 7.9. [78] Every countable product of Collins-Roscoe spaces has
the Collins-Roscoe property.
Since a compact space X is a Gul’ko compact space if and only if X
embeds into a Σs-product of real lines, we have:
Corollary 7.10. [38] Any Gul’ko compact space is a Collins-Roscoe space.
Any space X which has a weakly-point-finite T0-separating family of coz-
ero subsets can be condensed into some Σs-product of real lines. Thus, by
Proposition I.10.7 we have:
Corollary 7.11. [78] Suppose that X is a Lindelöf Σ-space and there
exists a weakly σ-point-finite T0-separating family of cozero subsets of X.
Then the space X has the Collins-Roscoe property and, in particular, it is
hereditarily metalindelöf.

Open problems and questions
Now we have a considerable amount of information about monotonically
monolithic spaces and Collins-Roscoe spaces. However, the unsolved prob-
lems are numerous. In this section we give the list of some of the more
intriguing open questions in this topic.
One of the central questions on D-spaces posed by E.K. van Douwen and
W. Pfeffer is whether every Lindelöf space is a D-space. As we have seen,
the concept of a D-space was studied a great deal ever since in almost every
context and Cp-theory was not an exception. The following question by A.V.
Arhangel’skii shows that the Cp-version of the question by E.K. van Douwen
and W. Pfeffer is also open.
Question 1.1. [28] Given a space X, suppose that Cp(X) is Lindelöf.
Must Cp(X) be a D-space?
For an arbitrary compact space we have that X is D and Cp,2n+1(X) is
hereditarily a D-space for any n ∈ ω. However, we do not know:
Question 1.2. [74],[76] Is it true that Cp(Cp(X)) is a D-space for any
compact space X?
If in addition X is a Corson compact space, we have that X is hered-
itarily a D space and Cp,2n+1(X) is hereditarily a D-space for any n ∈ ω.
Furthermore, Gul’ko proved that, for any Corson compact X, the odd iter-
ated function spaces are Lindelöf and the even ones are normal. This result
was strengthened by Sokolov [64] who established that all iterated function
spaces of a Corson compact space are Lindelöf. V.V. Tkachuk extracted from
Sokolov’s method the notion of a Sokolov space. It is known that for any
Sokolov space X with l∗(X) · t∗(X) ≤ ω the space Cp,n(X) is Sokolov and
l∗(Cp,n(X)) · t∗(Cp,n(X)) ≤ ω for any n ∈ ω [71]. Of course, any Corson
compact space X is Sokolov and satisfies l∗(X) · t∗(X) ≤ ω [71]. So, the
following question might have a chance for an affirmative answer.
Question 1.3. [74] Let X be a Corson compact space. Given any n ∈ ω,
the iterated function space Cp,n(X) is Sokolov and Lindelöf, but must Cp,2n(X)
be a D-space for any n ≥ 1?
A compact Sokolov space X need not be a Corson compact space, but
satisfies l∗(X) · t∗(X) ≤ ω [71]. So we can ask in a more general form.
65
66 OPEN PROBLEMS AND QUESTIONS
Question 1.4. [74] Let X be a Sokolov compact space. Given any n ∈ ω,
the iterated function space Cp,n(X) is Sokolov and Lindelöf, but must Cp,2n be
a D-space for any n > 1?
As we have seen, Sokolov compact spaces are a natural generalization of
Corson compact spaces. On the one hand it is natural to extend to the class
of Sokolov compact spaces the results obtained for Corson compact spaces.
On the other hand, Sokolov property has a strong relationship with the class
of D-spaces. In particular, all Sokolov spaces have countable extent so they
are Lindelöf whenever they have the D-property. It is not clear whether the
converse is true.
Question 1.5. [74] Let X be a Sokolov compact space. Must X be a
hereditarily D-space?
Question 1.6. [74] Suppose that X is a Sokolov Lindelöf space. Must X
be a D-space?
It is easy to see that the ordinal space ω1 embeds in a Σ-product of real
lines as a closed subspace, so Gul’ko’s results are applicable to prove that
Cp(ω1) is Lindelöf. R. Buzyakova generalized this result establishing that, for
any first countable countably compact subspace X of an ordinal, the space
Cp(X) is Lindelöf. It was proved in [16] that Cp(ω1) is a D-space. In [52]
and [74] this result was generalized by proving that: if X is a first countable
countably compact subspace of an ordinal, then Cp(X) is a D-space. In
addition, in [74], was established that if X is a first countable countably
compact subspace of an ordinal then X is a Sokolov space. As a consequence
Cp(X)ω, Cp(X
ω) and Cp,2n+1 for n ∈ ω are Lindelöf whenever X is a first
countable countably compact subspace of an ordinal.(see §6, Chapter I). The
questions below are inspired by these results and by the relationship between
Sokolov spaces and D-spaces.
Question 1.7. [74] Suppose that X is a first countable subspace of an
ordinal and ext(X) = ω. Must the space X be Sokolov?
Question 1.8. [74] Suppose that X is a first countable subspace of an
ordinal and ext(X) = ω. Must Cp(X) be a D-space?
Question 1.9. [74] Suppose that X is a Sokolov first countable space.
Then Cp(X) is Lindelöf and Sokolov, but must it be a D-space?
Any first countable countably compact subspace of an ordinal is a Sokolov
space. The space ω1 is Sokolov, first countable and countably compact; be-
sides, it embeds in a Σ-product of real lines. Therefore, it is a natural hy-
pothesis whether every first countable Sokolov space can be condensed into
a Σ-product of real lines. However, in [74] an example of a Sokolov first
countable countably compact space of cardinality c
+ which cannot be con-
densed into a Σ-product of real lines was constructed. So, the hypothesis
which follows are a natural change:
OPEN PROBLEMS AND QUESTIONS 67
Question 1.10. [74] Let X be the set of all countably cofinal ordinals in
ω2. Is it true in ZFC that X cannot be condensed (or, equivalently, embedded)
into a Σ-product of real lines?
A famous theorem of Gul’ko says that if X is compact and Cp(X) is
Lindelöf Σ then X is Corson compact, i.e., there exists an embedding of X
into a Σ-product of real lines. In [73] a general fact about Lindelöf Σ-spaces
was established which implies that if both spaces X and Cp(X) are Lindelöf
Σ then there is a rich family of retractions on the space X. As a consequence,
for any Tychonoff space X, if Cp(X) is Lindelöf Σ then X can be condensed
into a Σ-product of real lines. This gave an essential strengthening of Gul’ko’s
theorem.
We know that if Cp(X) is Lindelöf Σ then X has the Collins-Roscoe
property. Also, a compact space with the Collins-Roscoe property is a Corson
compact and hence can be embedded into a Σ-product of real lines. The next
question is open.
Question 1.11. [78] Let X be a Lindelöf Σ-space with the Collins-Roscoe
property. Is it true that X can be condensed into a Σ-product of real lines?
G. Gruenhage [35] asked if the Collins-Roscoe property is equivalent to
monotonic monolithicity, and suggested that Cp(X) for some Lindelöf Σ-space
X might be a place to look for an example distinguishing the two concepts.
V.V. Tkachuk [78] has shown that Cp(βD) does not have the Collins-Roscoe
property whenever D is an uncountable discrete space, but it is monotonically
monolithic. However, this question for compact spaces and Lindelöf Σ-spaces
remains open:
Question 1.12. [75] Suppose that X is a monotonically monolithic com-
pact space. Must X have the Collins-Roscoe property?
Question 1.13. [75] Suppose that X is a monotonically monolithic Lin-
delöf Σ-space. Must X have the Collins-Roscoe property?
It is known that if X is a compact monotonically ω-monolithic space
then X is monotonically monolithic. Since Lindelöf Σ-spaces are a natural
generalization of compact spaces, it is also natural to ask if this result can be
extended to the class of Lindelöf Σ-spaces.
Question 1.14. [75] Suppose that X is a monotonically ω-monolithic Lin-
delöf Σ-space. Must X be monotonically monolithic?
In [1] the problem of classifying monotonically monolithic generalized or-
dered spaces was proposed. It seems to be a difficult problem. The main
conjecture is that any monotonically monolithic compact linearly ordered
topological space is metrizable. Any space S ⊂ ω1 has countable tightness, so
it is monotonically ω-monolithic if and only if it is monotonically monolithic.
It was proved in [1] that a subspace S ⊂ ω1 is monotonically ω-monolithic if
68 OPEN PROBLEMS AND QUESTIONS
and only if it is not stationary, if and only if it is metrizable. The following
problem is open.
Problem 1.15. [1] Given an ordinal α, give a description of all mono-
tonically (ω-)monolithic subspaces of α.
It is well known that the Lindelöf Σ-property of Cp(X) usually has very
strong implications even if nothing is assumed about X. In this case, the
Lindelöf Σ-property of Cp(X) implies that Cp,2n(X) has the Collins-Roscoe
property for any n ∈ ω. But we do not know if in this case Cp(X) has itself
the Collins-Roscoe property. In the case when the network weight of X is
almost ω1 we know that the answer is positive, but in the general case we do
not know if Cp(X) is at least hereditarily metalindelöf or has the hereditary
D-property.
Question 1.16. [78] Suppose that Cp(X) is a Lindelöf Σ-space. Is it true
that Cp(X) has the Collins-Roscoe property?
Question 1.17. [78] Suppose that Cp(X) is a Lindelöf Σ-space. Is it true
that every subspace Y ⊂ Cp(X) is metalindelöf?
Question 1.18. [76], [78] Suppose that Cp(X) is a Lindelöf Σ-space. Must
Cp(X) itself be a hereditary D-space?
Recall that ω1 is a caliber of a space X if every point-countable family of
non-empty open subsets of X is countable. Clearly, if ω1 is a caliber of X then
any disjoint family of open sets in X is countable; that is, c(X) ≤ ω1. It can
be seen that sometimes being ω1 a caliber ofX implies very strong restrictions
on the topological structure of X or Cp(X). It is known that if a space X has
the Collins-Roscoe property and ω1 is a caliber of X then X is cosmic [78].
However, it is not known if this result holds when we only assume monotone
monolithicity of X, even when X is Lindelöf or has countable tightness.
Question 1.19. [1] Assume that X is a monotonically monolithic space
and ω1 is a caliber of X. Must X have a countable network?
Question 1.20. [1] Assume that X is a monotonically monolithic Lindelöf
space and ω1 is a caliber of X. Must X have a countable network?
Question 1.21. [1] Assume that X is a monotonically monolithic space,
t(X) ≤ ω, and ω1 is a caliber of X. Must X have a countable network?
In the presence of compactness the above result can be strengthened.
Indeed: If a compact space X is monotonically monolithic and ω1 is a caliber
of X, then X is metrizable [77]. It would be interesting to improve this
result as follows (recall that we do not know if monotone monolithicity and
the Collins-Roscoe property coincide in the class of compact spaces):
Question 1.22. [75] Suppose that X is a monotonically monolithic com-
pact space with c(X) ≤ ω. Must X be metrizable?
OPEN PROBLEMS AND QUESTIONS 69
Question 1.23. [78] Suppose that a compact space X has the Collins-
Roscoe property and c(X) = ω. Must X be metrizable?
Furthermore, in the presence of strong monotone monolithicity we can also
obtain strong restrictions over X. Indeed: If X is a strongly monotonically
monolithic space, ω1 is a caliber of X, and d(X) ≤ ω1, then X is metrizable.
Furthermore, under CH, if X is a strongly monotonically monolithic space
and ω1 is a caliber of X, then X is metrizable [1]. Of course, it would be
interesting improve these results by answering the following questions.
Question 1.24. [1] It is true in ZFC that any strongly monotonically
monolithic space with caliber ω1 is second countable?
Question 1.25. [1] Assume that X is a strongly monotonically monolithic
Lindelöf space and ω1 is a caliber of X. Is it true in ZFC that X must be
second countable?
It is still an open question whether every strong Collins-Roscoe space has
a point-countable base. The Corollary to Theorem 5 of [45] establishes that
every strong Collins-Roscoe space X has a dense subspace Y with a point-
countable base. As is noted in [75] it is possible to extract from the proof
of Theorem 5 of [45] that Y actually has a point-countable external base B.
In [75] it was proved that if X is a Collins-Roscoe space, then every left-
separated subspace Y ⊂ X has a point-countable open expansion. It is not
known if these results can be strengthened as follows, when compactness of
the space X is assumed.
Question 1.26. [75] Suppose that X is a Collins-Roscoe compact space.
Must X have a dense metrizable subspace?
Question 1.27. [75] Suppose that X is a monotonically monolithic com-
pact space. Must X have a dense metrizable subspace?
For a maximal almost disjoint family A of subsets of ω, under CH the
space Cp(Ψ(A)) is a hereditarily D-space. Recall that a maximal almost
disjoint family A is a Mrówka mad family if the one-point compactification
of Ψ(A) coincides with its Stone-Čech compactification. If we do not as-
sume CH and A is a Mrówka mad family, it is not difficult to see, using
a slight modification of Proposition II.5.8, that Cp(Ψ(A)) is monotonically
<κ-monolithic when nw(Cp(Ψ(A))) = nw(Ψ(A)) = |A|. Hence, in this case
Cp(Ψ(A)) is a hereditarily D-space. Furthermore, it was proved in [43] that if
A is an infinite maximal almost disjoint family on ω, then the space Cp(Ψ(A))
is not normal, and its extent and Lindelöf number are all equal to |A|.
We think that the following question might have a chance for an affirma-
tive answer.
Question 1.28. [62] Is it true in ZFC that for every almost disjoint
family A, that space Cp(Ψ(A)) is a hereditarily D-space?
70 OPEN PROBLEMS AND QUESTIONS
If κ is an infinite cardinal number of countable cofinality, then κ is a σ-
compact space, so Cp([0, κ)) is monotonically monolithic and hence a hered-
itarily D-space. If κ is regular we know that Cp([0, κ)) is monotonically
<κ-monolithic where nw(Cp([0, κ))) = κ. Hence, in this case Cp([0, κ)) is
also a hereditarily D-space. However, we do not know if this result holds for
singular cardinals.
Question 1.29. [62] Let λ be a singular cardinal of uncountable cofinality.
Is it true that Cp([0, λ)) is a D-space (resp., a hereditarily D-space)?
It is a classical result that any product (σ-product or Σ-product) of Lin-
delöf Σ-spaces is stable. In [61] the concept of monotonically stable space
was introduced and it was proved that any product (σ-product or Σ-product)
of cosmic spaces is monotonically stable. However, for Lindelöf Σ-spaces it
was only proved that any product of such spaces is monotonically stable.
Furthermore, for an infinite cardinal λ, if κ = λ+ and Y is a Σκ-product of
a family Lindelöf Σ-spaces, then Y is monotonically λ-stable. Therefore, the
next question arises in a natural form.
Question 1.30. [61] Is it true that every σ-product or Σ-product of an
arbitrary family of Lindelöf Σ-spaces is monotonically stable?
Any σ-product of compact spaces is σ-compact and hence, monotonically
stable. In a more general form, any Σs-product of compact spaces is Lindelöf
Σ and thus, monotonically stable. But we do not know:
Question 1.31. [61] Is it true that every Σ-product of an arbitrary family
of compact spaces is monotonically stable?
Finally, the definition of monotonically stable is in terms of real-valued
continuous functions and the pointwise convergence topology, so it would be
interesting to obtain some intrinsec characterization of such concept.
Problem 1.32. [61] Find an internal characterization of monotone sta-
bility.
Bibliography
[1] O. Alas, V. Tkachuk, and R. Wilson, A broader context for monotonically monolithic
spaces, Acta Math. Hungarica 125:4 (2009), 369-385.
[2] K. Alster, Some remarks on Eberlein compacts, Fund. Math. 104 (1979), 43-46.
[3] A.V. Arhangel’skii, Topological Function Spaces, Kluwer Acad. Publ., Dordrecht, 1992.
[4] A.V. Arhangel’skii, Continuous mappings, factorization theorems and function spaces,
Trudy Mosk. Mat. Obsch. 47 (1984), 3-21 (in Russian).
[5] A.V. Arhangel’skii, Topological properties of function spaces: duality theorems, Soviet
Math. Dokl. 27 (1983), 470-473 (Translated from the Russian).
[6] A.V. Arhangel’skii and R. Buzyakova, Addition theorems and D-spaces, Comment.
Mat. Univ. Car. 43 (2002), 653-663.
[7] A.V. Arhangel’skii and M. Tkachenko, Topological groups and related structures, At-
lantis Press-World Scientific., 2008.
[8] L. Aurichi, D-spaces, topological games, and selection principles, Topology Proc. 36
(2010), 107-122.
[9] L. Aurichi, L. Junqueira, P. Larson, D-spaces, irreducibility, and trees, Topology Proc.
35 (2010), 73-82.
[10] D. Baturov, On subspaces of function spaces, Vestn. Moskov. Univ. Ser. I Mat. Mech.
4 (1987), 66-69;
[11] H. Bennett , D. Lutzer , and M. Matveev, The monotone Lindelöf property and sepa-
rability in ordered spaces, Topology Appl. 151 (2005), 180-186.
[12] C.R. Borges and A. Wehrly, A study of D-spaces, Topology Proc. 16 (1991), 7-15.
[13] C.R. Borges and A. Wehrly, Another study of D-spaces, Questions and Answers in
Gen. Topology 14 (1996), 73-76.
[14] D. Burke, Covering properties, in the Handbook of Set-theoretic Topology, K. Kunen
and J.E. Vaughan, eds., North-Holland, Amsterdam, 1984, 347-442.
[15] D. Burke, Weak bases and D-spaces, Comment. Mat. Univ. Car. 48 (2007), 281-289.
[16] R. Buzyakova, Hereditary D-property of function spaces over compacta, Proc. Amer.
Math. Soc. 132 (11) (2004), 3433-3439.
[17] R. Buzyakova, How sensitive is Cp(X,Y ) to changes in X and/or Y ?, Comment.Math.
Univ. Carolin. 49 (4) (2008), 657-665.
[18] R. Buzyakova, On D-property of strong Σ-spaces, Comment. Mat. Univ. Car. 43
(2012), 493-495.
[19] R. Buzyakova, In search for Lindelöf Cp’s, Comment. Math. Univ. Carolin. 45 (1)
(2004), 145-151.
[20] J. Ceder, Some generalizations of metric spaces, Pacific J. Math. 11 (1961), 105-125.
[21] J. Chaber, On point-countable collections and monotonic properties, Fund. Math. 94:3
(1997), 209-219.
[22] P.J. Collins and A.W. Roscoe, Criteria for metrisability, Proc. Amer. Math. Soc. 90
(4) (1984), 631-640.
[23] E.K. van Douwen, A technique for constructing honest locally compact submetrizable
examples, Topology Appl. 47 (3) (1992), 179-201.
71
72 BIBLIOGRAPHY
[24] E.K. van Douwen and H. Wicke, A real, weird topology on the reals, Houston J. Math.
3 (1) (1977), 141-152.
[25] E.K. van Douwen and W.F. Pfeffer, Some properties of the Sorgenfrey line and related
spaces, Pacific J. Math. 81 (1979), 371-377.
[26] A. Dow, H. Junnila, J. Pelant, Coverings, networks, and weak topologies, Mathematika
53 (2006), 287-320.
[27] A. Dow, H. Junnila, J. Pelant, Weak covering properties of weak topologies, Proc.
London Math. Soc. 75:3 (1997), 349-368.
[28] T. Eisworth, On D-spaces, in: Open Problems in Topology II, E. Pearl, ed., Elsevier,
Amsterdam, 2007, 129-134.
[29] R. Engelking, General Topology, second edition, Sigma Series in Pure Mathematics,
6, Heldermann, Berlin, 1989.
[30] W. Fleissner and A. Stanley, D-spaces, Topology Appl. 114 (2001), 261-271.
[31] P.M. Gartside and P.J. Moody, A note on proto-metrisable spaces, Topology Appl. 52
(1993), 1-9.
[32] L. Gillman and M. Jerison, Rings of continuous functions, Princeton, 1960.
[33] C. Good, R. Knight, and I. Stares, Monotone countable paracompactness, Topology
Appl. 101 (2000), 281-298.
[34] A. Grothendieck, Critères de compacité dans les espaces fonctionnels généraux, Amer.
J. Math. 74 (1952), 164-186.
[35] G. Gruenhage, A note on D-spaces, Topology Appl. 153 (2006), 2229-2240.
[36] G. Gruenhage, A survey of D-spaces, Contemporary Math. 553 (2011), 13-28.
[37] G. Gruenhage, Covering properties on X2, W -sets, and compact subsets of Σ-products,
Topology Appl. 17 (1984), 287-304.
[38] G. Gruenhage, Monotonically monolithic spaces, Corson compacts and D-spaces,
Topology Appl. 159 (2012), 1559-1564.
[39] H. Guo and H.J.K. Junnila, D-spaces and thick covers, Topology Appl. 158 (2011),
2111-2121.
[40] R. E. Hodel, Cardinal functions I. Handbook of Set-Theoretic Topology, K. Kunen and
J. E. Vaughan, eds., North-Holland, 1984, 1-61.
[41] M. Hrušák and J. T. Moore, Twenty problems in set theoretic topology, in: Open
problems in topology II, E. Pearl, ed., Elsevier, Amsterdam, 2007, 111-114.
[42] I. Juhász, Cardinal Functions in Topology—Ten Years Later, Mathematical Centre
Tracts, vol. 123 Mathematisch Centrum, Amsterdam, 1980.
[43] M. Hrušák P.J. Szeptycki, and Á. Tamariz-Mascarúa Spaces of continuous functions
defined on Mrówka spaces, Topology and its Applications 148 (2005), 239-252.
[44] F. Lin and S. Lin, Sequence covering maps in generalized metric spaces, arXiv:
1106.3806v1 [Math.GN], 20 June 2011.
[45] P. J. Moody, G. M. Reed, A. W. Roscoe, P.J. Collins, A lattice of conditions on
topological spaces II, Fund. Math. 138:2 (1991), 69-81.
[46] S. Lin, A note on D-spaces, Comment. Math. Univ. Carolin. 47 (2006), 313-316.
[47] P. Nyikos, D-spaces, trees, and an answer to a problem of Buzyakova, Topology Proc.
38 (2011), 361-373.
[48] O.G. Okunev, On Lindelöf property of spaces of continuous functions over a Tychonoff
space and its subspaces, Comment. Math. Univ. Carolin. 50 (4) (2009), 629-635.
[49] O.G. Okunev, On Lindelöf Σ-spaces of continuous functions in the pointwise topology,
Topology Appl. 49 (2) (1993), 149-166.
[50] C. Pan, Monotonically CP spaces, Questions and Answers in Gen. Topology 15 (1997),
25-32.
[51] L.-Xue Peng, A note on D-spaces and infinite unions, Topology Appl. 154 (2007),
2223-2227.
BIBLIOGRAPHY 73
[52] L.-Xue Peng, The D-property of some Lindelöf spaces and related conclusions, Topol-
ogy Appl. 154 (2007), 469-475.
[53] L.-Xue Peng, The D-property which relates to certain covering properties, Topology
Appl. 159 (2012), 869-876.
[54] L.-Xue Peng, On spaces which are D, linearly D and transitively D, Topology Appl.
157:2 (2010), 278-394.
[55] L.-Xue Peng, On weakly monotonically monolithic spaces, Comment. Math. Univ.
Carolin. 51:1 (2010), 133-142.
[56] L.-X. Peng, H. Li, The D-property of monotone covering properties and related con-
clusions, Topology Appl.159:15 (2012), 3274-3281.
[57] S. Popvassilev, Base-cover paracompactness, Proc. Amer. Math. Soc. 132 (2004), 3121-
3130.
[58] S. Popvassilev, ω1+1 is not monotonically countably metacompact, Questions Answers
Gen. Topology 27 (2009), 133-135.
[59] J.E. Porter, Generalizations of totally paracompact spaces, Dissertation, Auburn Uni-
versity, 2000.
[60] E.A. Reznichenko, On convex and compact subsets of functional and locally convex
spaces, Ph.D. Dissertation, Moscow University, Moscow (1992), p. 114.
[61] R. Rojas-Hernández, A note on monotone monolithicity, submitted.
[62] R. Rojas-Hernández, Á. Tamariz-Mascarúa, D-property, monotone monolithicity and
function spaces, Topology Appl. 159 (2012), 3379-3391.
[63] D.B. Shakhmatov, On pseudocompact spaces with a point-countable base, Soviet Math.
Dokl. 30 (1984), 747-751.
[64] G.A. Sokolov, Lindelöf spaces of continuous functions (in Russian), Matem. Zametki
39:6 (1986), 887-894.
[65] G.A. Sokolov, On some class of compact spaces lying in Σ-products, Comment. Math.
Univ. Carolinae 25:2 (1984), 219-231.
[66] D.T. Soukup, P.J. Szeptycki, A counterexample in the theory of D-spaces, Topology
Appl. 159 (2012), 2669-2678.
[67] D.T. Soukup, X. Yuming, The Collins-Roscoe mechanism and D-spaces, Acta Math.
Hungar. 131:3 (2011), 275-284.
[68] R.M. Stephenson, Jr., Initially κ-compact and related spaces, in the Handbook of Set-
theoretic Topology, K. Kunen and J.E. Vaughan, eds., North-Holland, Amsterdam,
1984, 603-632.
[69] H. Teng, S. Xia, and S. Lin, Closed images of some generalized countably compact
spaces, Chinese Ann. Math. Ser. A 10 (1989), 554-558.
[70] V.V. Tkachuk, A Cp-Theory Problem Book: Topological and Function Spaces,
Springer, New York, 2011.
[71] V.V. Tkachuk, A nice class extracted from Cp-theory, Comment. Math. Univ. Carolin.
46:3 (2005), 503-5013.
[72] V.V. Tkachuk, Behaviour of the Lindelöf Σ-property in iterated function spaces, Topol-
ogy Appl. 107 (2000), 297-305.
[73] V.V. Tkachuk, Condensing function spaces into Σ-products of real lines, Houston J.
Math. 33 (2007), 209-228.
[74] V.V. Tkachuk, Countably compact first countable subspaces of ordinals have the
Sokolov property, Quaestiones Mathematicae 34 (2011), 225-234.
[75] V.V. Tkachuk, Lifting the Collins-Roscoe property by condensations, Topology Proc.
42 (2012), 1-15.
[76] V.V. Tkachuk, Lindelöf Σ-spaces: an omnipresent class, RACSAM 104:2 (2010), 221-
224.
74 BIBLIOGRAPHY
[77] V.V. Tkachuk, Monolithic spaces and D-spaces revisited, Topology Appl. 156 (2009),
840-846.
[78] V.V. Tkachuk, The Collins-Roscoe property and its applications in the theory of func-
tion spaces, Topology Appl. 159 (2012), 1529-1535.
[79] H. Zhang and W.-X. Shi, A note on D-spaces, Topology Appl. 159 (2012) 248-252.
[80] Y. Zuoming a, Y. Ziqiub, On spaces with k-in-countable bases or weak bases, Topology
Appl. 159 (2012), 3545-3549.