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Revista Integración
Escuela de Matemáticas
Universidad Industrial de Santander
Vol. 38, N◦2, 2020, pág. 93–102
Some topological properties of C-normality
I. E. Soberano-Gonzáleza, G. Delgadillo-Piñónb,
R. Rojas-Hernándezc
a, b Universidad Juárez Autónoma de Tabasco, División Académica de Ciencias
Básicas, Tabasco, México.
E-mail: aisoberanogonzalez@gmail.com,bgerardo.delgadillo@ujat.mx.
cUniversidad Michoacana de San Nicolás de Hidalgo, Facultad de Ciencias Físico
Matemáticas, Michoacán, México. E-mail: satzchen@yahoo.com.
Abstract. A topological space Xis C-normal if there exists a bijective
function f:X→Y, for some normal space Y, such that the restriction
f↾C:C→f(C)is a homeomorphism for each compact C⊂X. The purpose
of this work is to extend the known classes of C-normal spaces and clarify the
behavior of C-normality under several usual topological operations; in partic-
ular, it is proved that C-normality is not preserved under closed subspaces,
unions, continuous and closed images, and inverse images under perfect func-
tions. These results are used to answer some questions raised in [1], [2] and
[6].
Keywords: Normality, local compactness, epi-normality, compactness.
MSC2010: 54D15, 54D20, 54B05, 54B10, 54B15.
Algunas propiedades topológicas de la C-normalidad
Resumen. Un espacio topológico Xes C-normal si existe una función bi-
yectiva f:X→Y, para algún espacio normal Y, tal que la restricción
f↾C:C→f(C)es un homeomorfismo para cada compacto C⊂X. El
propósito de este trabajo es extender las clases conocidas de los espacios
C-normales y aclarar el comportamiento de C-normalidad bajo varias opera-
ciones topológicas habituales; en particular, se demuestra que la normalidad
Cno se conserva bajo subespacios cerrados, uniones, imágenes continuas y
cerradas e imágenes inversas bajo funciones perfectas. Estos resultados se uti-
lizan para responder algunas preguntas planteadas en [1], [2] y [6].
Palabras clave: Normalidad, compacidad local, epi-normalidad, compacidad.
0Corresponding author.
Received: 17 December 2019, Accepted: 10 March 2020.
To cite this article: I. E. Soberano-González, G. Delgadillo-Piñón and R. Rojas-Hernández, Some
topological properties of C-normality, Rev. Integr. temas mat. 38 (2020), No. 2, 93-102.
doi: 10.18273/revint.v38n2-2020002
93
94 I.E. Soberano-González, G. Delgadillo-Piñón & R. Rojas-Hernández
1. Introduction
In 2012 Arhangel’skii proposed the study in General Topology of two variants of nor-
mality; C-normality and epi-normality. Years later AlZahrani and Kalantan published a
study of the behavior of these two topological properties and their relations with other
normal-type properties (see [1],[6]).
At the beginning of this work we present a systematic study of the classes C-Pand epi-
Pof topological spaces. These classes are defined in a similar way to C-normality and
epi-normality, but considering an arbitrary topological property Pinstead of normality.
We show that the classes C-Pand epi-Pare hereditary (additive or productive) when P
is hereditary (additive or productive, respectly). Then we apply these results to study
C-normal spaces; we extend the known classes of C-normal spaces by showing that
they include products of locally compact spaces and locally Lindelöf spaces. We also
describe some specific examples. In [6] Saeed showed the existence of a Tychonoff space
which is not C-normal; we use some spaces associated with such example to prove that
C-normality is not preserved under closed subspaces, unions of subspaces, continuous
and closed images, and perfect preimages. This shows that the categorical behavior of
C-normality is very different from normality’s categorical behavior, and answers some
questions posed in [1]. We conclude the work comparing some characteristics of C-
normality and epi-normality.
2. Notation
Throughout the text all spaces under consideration will be assumed to be Hausdorff. The
symbol ωrepresents the first infinite ordinal and ω1is the first uncountable ordinal. The
continuum is denoted by c. The set of natural numbers is denoted by Nand the symbol
Rstands for the set of real numbers.
We say that a space Xis a k-space if a set U⊂Xis open if, an only if, U∩Cis open
in Cfor every compact C⊂X. The space Xis locally Lindelöf if for each point xin
Xthere is a neighborhood Uof xwhich is Lindelöf. The space Xis Urysohn if for each
pair of different points x, y ∈Xthere exist open sets U, V ⊂Xsatisfying x∈U,y∈V
and U∩V=∅.
Given a space X, we denote as A(X)the Alexandroff duplicate X∪X′of X, where X′
is a disjoint copy of Xand there exist a bijective assignment x7→ x′from Xonto X′.
Given a set U⊂Xwe choose U′={x′}x∈U. The topology of A(X)is defined as follows.
All points of X′are isolated and a point x∈Xhas as a basis of open neighborhoods the
family of all sets of the form U∪U′\ {x′}, where Uis a open neighborhood of xin X.
All non stated concepts and notation can be understood as in [5].
3. The classes of epi-Pand C-Pspaces
The following notions describe two different ways in which we can extend the class of all
topological spaces satisfying a given property.
Definition 3.1. Let Pbe a topological property.
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Some topological properties of C-normality 95
A topological space Xis called epi-Pif there exists a bijective continuous function
f:X→Yfor some space Ywhich satisfies P.
A topological space Xis C-Pif there exists a bijective function f:X→Y, where
Yhas property P, and f↾C:C→f(C)is a homeomorphism for each compact
C⊂X.
Given a topological property P, since every bijective continuous function defined on a
compact Hausdorff space is a homeomorphism onto its image, all epi-Pspaces are C-P.
The other implication is not always true, for example when Pcoincides with normality
(see Example 6.5). The following result gives us a condition under which these two
notions are equivalent; the proof follows since for a k-space Xa function f:X→Yis
continuous if, and only if, f↾Cis continuous for each compact C⊂X(see [5, Theorem
3.3.21]).
Proposition 3.2. If Pis a topological property and Xis a k-space, then Xis C-Pif,
and only if, Xis epi-P.
The classes C-Pand epi-Pcan coincide, for example when a space Xsatisfies Pif, and
only if, every compact subset of Xis metrizable.
If a topological property Pimplies another topological property Q, then all epi-P(C-P)
spaces are epi-Q(C-Q). Besides, if Pand Qare different properties, the class of epi-P
spaces and the class of epi-Qspaces can coincide; for example, when Qis the class of
epi-Pspaces the class of epi-Pspaces coincides with the class of epi-Qspaces. Similarly,
the classes C-Pand C-Qcan coincide, as we will show now.
Theorem 3.3. If Pis a topological property, the class of C-Pspaces and the class of
C-(C-P)spaces coincide.
Proof. It is sufficient to prove that every C-(C-P)space is C-P. Suppose that there
exists a bijective function f:X→Y, where Yis C-Pand f↾C:C→f(C)is a
homeomorphism for each compact subspace C⊂X; we shall prove that Xis C-P. As Y
is C-P, there exists a space Zwith property Pand a bijective function g:Y→Zsuch
that g↾D:D→g(D)is a homeomorphism for each compact subspace D⊂Y. We claim
that g◦fwitnesses that Xis C-P. Indeed, let C⊂Xcompact. Since f↾C:C→f(C)
is a homeomorphism, the space D=f(C)is compact. It follows that g↾D:D→g(D)is
a homeomorphism. Thus (g◦f)↾C= (g↾D)◦(f↾C) : C→g◦f(C)is a homeomorphism
and, since Zhas property P, we conclude that Xis C-P.□
✓
✓
✓
In what follows we will analyze some properties of the classes epi-Pand C-Pinherited
from the property P.
Theorem 3.4. If a property Pis hereditary, then the classes C-Pand epi-Pare closed
under arbitrary subspaces.
Proof. We will show the case of C-Pspaces; the proof for the epi-Pspaces is similar. Let
Abe a subset of X. As Xis a C-Pspace, there exists a bijective function f:X→Y,
where Yhas property P, such that f↾C:C→f(C)is a homeomorphism for each
Vol. 38, No. 2, 2020]
96 I.E. Soberano-González, G. Delgadillo-Piñón & R. Rojas-Hernández
compact subspace C⊂X. Since the property Pis hereditary, the space f(A)has
property P. It is clear that f↾A:A→f(A)is bijective. Since any compact subspace
of Ais compact in X, the restriction (f↾A)↾C=f↾Cis a homeomorphism for each
compact subspace C⊂A. Thus Ais C-P.□
✓
✓
✓
Theorem 3.5. If κis a cardinal and Pis a κ-productive property, then the classes C-P
and epi-Pare closed under products of κ-factors.
Proof. We will prove the result for the class C-P; the case of the class epi-Pis similar.
Let {Xs}s∈Sbe a family of C-Pspaces where Shas cardinality κ. For each s∈Slet
fs:Xs→Ysbe a bijective function for some Yswith property Psuch that fs↾Cs:
Cs→fs(Cs)is a homeomorphism for each compact subspace Cs⊂Xs. Note that
f=∏s∈Sfs:∏s∈SXs→∏s∈SYsis bijective. Besides, as Pis a κ-productive property,
it follows that ∏s∈SYshas property P. Given a compact set C⊂∏s∈SXs, notice that
D=∏s∈Sπs(C)is compact, and so the function f↾D=∏s∈Sfs↾πs(C)=D→f(D)is
a homeomorphism; consequently, f↾C:C→f(C)also is a homeomorphism. Thus, the
product ∏s∈SXsis C-P.□
✓
✓
✓
Theorem 3.6. If Pis a κ-aditive property, then the classes C-Pand epi-Pare closed
under disjoint sums of κ-factors.
Proof. We will prove the result for the class of C-Pspaces, the other case is similar. Let
{Xs}s∈Sbe a family of spaces C-Pwhere |S|=κ. For each s∈S, let fs:Xs→Ysbe
a bijective function for some space Yswith property Psuch that fs↾Cs:Cs→fs(Cs)is
a homeomorphism for each compact subspace Cs⊂Xs. As Pis a κ-additive property,
it follows that ⊕s∈SYshas property P. Besides, the function ⊕s∈Sfs:⊕s∈SXs→
⊕s∈SYsis bijective. Now let C⊂⊕s∈SXsbe a compact space, then the set S0=
{s∈S:C∩Xs6=∅} is finite and Cs=C∩Xsis compact for each s∈S0. Then
(⊕s∈Sfs)↾C=⊕s∈S0fs↾Csis a homeomorphism, because fs↾Csis a homeomorphism
for each s∈S0. Thus, the disjoint sum ⊕s∈SXsis C-P.□
✓
✓
✓
By an argument similar to the one used in the proof of Theorem 3.5 we can prove the
following result.
Proposition 3.7. Consider two properties Pand Qsuch that X×Yhas Pwhen Xhas
Pand Yhas Q. Then X×Yis C-Pwhen Xis C-Pand Yis C-Q.
Theorem 3.8. Let Pbe a property preserved under Alexandroff duplicates; then A(X)is
C-P(epi-P) whenever Xis C-P(epi-P).
Proof. We will show the case of C-Pspaces; the case for the epi-Pspaces is similar. Let
Xbe a C-Pspace; then, there exists a space Ywith property Pand a bijective function
f:X→Ysuch that f↾C:C→f(C)is a homeomorphism for each compact subspace
C⊂X. Consider the Alexandroff duplicated A(X)and A(Y)of Xand Y, respectly.
Since Yhas P, the space A(Y)also has P. Define F:A(X)→A(Y)by F(x) = f(x)
and F(x′) = f(x)′for each x∈X, the natural function induced by f. Notice that F
is a bijective function. Let C⊂A(X)be a compact subspace. We shall prove that
F↾C:C→F(C)is a homeomorphism. Let p:A(X)→Xbe the function given by
p(x) = p(x′) = x, for each x∈X. Observe that pis continuous. For the compact set
[Revista Integración
Some topological properties of C-normality 97
D=p(C)we have that g=f↾Dis bijective and continuous. It is easy to verify that
the natural function G:A(D)→A(g(D)) induced by g, given by G(x) = g(x)and
G(x′) = g(x)′for each x∈D, also is bijective and continuous. As A(D)is compact, the
function Gis a homeomorphism. We know that C⊂A(D)⊂A(X), thus F↾C=G↾C
also is a homeomorphism. □
✓
✓
✓
We consider now the following well known construction. Let Xbe an arbitrary space.
Take kX =X. Define a topology on kX as follows. A set of kX is open if, and only
if, its intersection with any compact subspace Cof Xis open in C. Then the space
kX endowed with this topology is a k-space, has exactly the same compact subspaces
that X, and induces the same topology that Xon these compact subspaces. From these
observations it is easy to conclude the following.
Proposition 3.9. Let Pbe a topological property. A space Xis C-Pif, and only if, kX is C-P.
4. C-normal spaces
In this text we will be particularly interested in C-normality and some related properties.
Notice that all epi-normal spaces, all C-compact spaces and all C-metrizable spaces are
C-normal. We will provide another classes of spaces which are C-normal.
As is stated in Exercise 3.3.D from [5], every locally compact space is epi-compact, so we
can apply Theorem 3.5 to obtain the following corollary.
Corollary 4.1. If {Xs}s∈Sis a family of locally compact spaces, then the product ∏s∈SXs
is epi-compact.
Example 4.2. The space of real numbers Ris locally compact, because of Corollary 4.1
the product RSis C-normal, for any set S. Moreover, if Sis the Sorgenfrey line, then
SSadmits a bijective continuous function onto RS, so we can apply Theorem 3.3 to see
that SSis C-normal. However, RSis not normal when the set Sis not countable (see [5,
Exercise 2.3.E]) and SSis not normal when Shas at least two elements (see [5, Example
2.3.12]).
Now we will deal with a notion more general than locally compactness, local Lindelöfness,
in order to get more examples of C-normal spaces.
Theorem 4.3. If Xis regular and locally Lindelöf, then Xis epi-Lindelöf.
Proof. We must prove that Xadmits a bijective and continuous function onto a Lindelöf
space. Let Y=X∪ {y}where y /∈X. We define a topology in Yin the following way.
The topology of Yis the minimal topology on Ywhich satisfies the following conditions:
1. It contains the topology of X.
2. It contains each set U⊂Ysuch that y∈Uand whose complement Y\U, is closed
in Xand has a neighborhood in Xwhose closure in Xhas the Lindelöf property.
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98 I.E. Soberano-González, G. Delgadillo-Piñón & R. Rojas-Hernández
As Xis regular and locally Lindelöf, the space Yis T1. We will verify now that Yis
regular. Given A⊂X, along this proof Aalways refers to the closure of Ain X. Take
a point x∈Yand a neighborhood Uof xin Y. If x6=y, since Xis regular, we can
suppose that U⊂Xand Uis Lindelöf. By the regularity of X, there exists an open
neighborhood Vof xsuch that x∈V⊂V⊂U. Notice that Vis closed in Y. If x=y
we can suppose that F=Y\Uis closed in Xand has a neighborhood Vin Xwhose
closure Vin Xhas the Lindelöf property. As Vis normal, there exists an open set Win
Xsuch that F⊂W⊂W⊂V⊂V. It follows that Wis closed in Y, and if O=Y\W,
then y∈O⊂ {y} ∪ O\ {y} ⊂ Y\W⊂Y\F=U, where {y} ∪ O\ {y}is the closure of
Oin Y. Thus, the space Yis regular.
It is easy to verify that Yis Lindelöf. Fix x∈Xand consider the space Zwhich is
obtained from Yidentifying the points xand y, and let q:Y→Zbe the quotient
function associated with this identification. As Yis normal and qonly identifies a closed
set, the space Zis regular. Since qis continuous, the space Zis Lindelöf. Finally, is
clear that the function q↾X:X→Zis bijective, and hence this function witnesses that
Xis epi-Lindelöf. □
✓
✓
✓
We now describe some examples of locally Lindelöf spaces, and hence C-normal spaces,
which are neither locally compact nor normal.
Example 4.4. Let Xbe a locally compact not normal space and let Ybe a Lindelöf not
locally compact space. Consider the space X×Y. Note that X×Yis not normal, because
it has a closed subspace homeomorphic to Xwhich is not normal. Observe that X×Y
is not locally compact, because it has a closed subspace homeomorphic to Ywhich is not
locally compact. However, the product X×Yis locally Lindelöf, because the product of a
compact space and a Lindelöf space is always Lindelöf. Thus, Theorem 4.3 implies that
X×Yis C-normal. As a particular case, we can take Xas the deleted Tychonoff plank
and Yas the Sorgenfrey line.
Example 4.5. Consider the following variant of a Ψ-space. Let Abe a maximal family
of uncountable subsets of ω1such that A∩Bis countable for each A, B ∈ A. It is easy
to deduce from the maximality of Athat |A| ≥ ω2. Consider the space Ψω1(A) = ω1∪ A,
where each point in ω1is isolated and every A∈ A has as a basis of open neighborhoods
the family {A\C:C∈[ω1]<ω1}. It fol lows immediately from the definition that Ψω1(A)
is locally Lindelöf and not locally compact.
We will prove that Ψω1(A)is not normal. Suppose on the contrary, that Ψω1(A)is
normal. Let {Aα}α<ω1be a partition of Ain nonempty subsets and fix Aα∈ Aαfor
each α < ω1. Given α < ω1, because of the normality of Ψω1(A)we can choose an
uncountable subset Bαof ω1such that the subsets Aα∪Bαand (A \ Aα)∪(ω1\Bα)
form a partition of Ψω1(A)in open sets. We will construct a subset {xα}α<ω1of ω1
recursively as follows. Fix x0∈A0, and if {xα}α<β is defined for some β < ω1, fix
xβ∈Aβ\∪α<β ({xα} ∪ Bα). Consider the uncountable set B={xα}α<ω1; then the
maximality of Aimplies that A∩Bis uncountable for some A∈ A. We know that A∈ Aγ
for some γ < ω1. Since {A} ∪ Bγis open, we must have that (A∩B)\Bγ⊂A\Bγ
is countable, and hence (A∩B)\Bγ⊂ {xα}α<β for some γ < β < ω1. Since A∩Bis
uncountable, we can suppose that βis such that xβ∈A∩B. However, the construction
implies that xβ6∈ Bγ, which is not possible. Thus, the space Ψω1(A)is not normal.
[Revista Integración
Some topological properties of C-normality 99
Question 4.6. Is there a locally normal regular space Xwhich is not C-normal?
Proposition 4.7. If any countable subspace of Xis discrete, then Xis C-normal.
Proof. By [1, Corollary 1.4] it is sufficient to verify that all compact subsets of Xare
finite; namely, under such conditions any bijection onto a discrete space witnesses the
C-normality of X. Let A⊂Xinfinite. Take B⊂Ainfinite and countable. Note that B
is closed in Xbecause B∪ {x}is discrete for each x∈X. As Bis closed, discrete and
infinite, it follows that Ais not compact. □
✓
✓
✓
We now describe an example of a space in which all countable subsets are discrete,
and hence a C-normal space, but which is not normal. This example was obtained by
Shakhmatov (see [3, Example 1.2.5]).
Example 4.8. Let Icbe the Tychonoff cube of weight c. Let
ΣIc={x∈Ic:|{α < c:πα(x)6= 0}| ≤ ω} ⊂ Ic.
Observe that |ΣIc|=c, and take an enumeration {xα}α<cof the elements of ΣIcwhere
each element appears c-many times. Moreover, take an enumeration {Aα}α<cof the
elements from [c]≤ωwhere each element appears c-many times. For each α < cdefine a
point yα∈Icby:
πβ(yα) =
πβ(xα),if β≤α;
1,if β > α, β ∈Aα;
0,if β > α, β 6∈ Aα.
As it is proved in [3, Example 1.2.5], the space Y={yα}α<c⊂Icis dense in Ic,
pseudocompact, and every countable subset of Yis discrete. As Yis pseudocompact but
not countably compact, we conclude from [5, Theorem 3.10.21] that Yis not normal.
5. Operations with C-normal spaces
In [6] Saeed showed the existence of a Tychonoff space which is not C-normal. Such
example is constructed as follows: Let 2ω1be the Cantor cube of size ω1; the product of
ω1-many copies of the discrete two points space. Now consider the subspace
Σ2ω1={x∈2ω1:|x−1(1)| ≤ ω} ⊂ 2ω1.
Then the product 2ω1×Σ2ω1is Tychonoff but not C-normal (see [6, Example 8]). This
example provides us a compact space and a normal space whose product is not C-normal,
so C-normality is not a productive property. However, we still do not know the answer
to the following question.
Question 5.1. Is there a C-normal space Xsuch that its square is not C-normal?
We know that normality is preserved under closed subspaces and closed continuous im-
ages. In the following examples we will show that C-normality is not necessary preserved
in these cases.
Example 5.2. There exists an epi-compact space containing a closed subspace which is
not C-normal.
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100 I.E. Soberano-González, G. Delgadillo-Piñón & R. Rojas-Hernández
Proof. Consider the cartesian product Y= 2ω1×2ω1endowed with the product topology,
and the cartesian product X= 2ω1×2ω1endowed with the topology obtained from the
product topology by adding 2ω1×Σ2ω1and its complement as open sets. Notice that X
is epi-normal; indeed, the identity function id :X→Yis continuous and Yis compact.
It is clear that 2ω1×Σ2ω1is closed in X. Moreover, the topology on 2ω1×Σ2ω1inherited
from Xcoincides with the topology inherited from Y. Thus, 2ω1×Σ2ω1is a closed
subspace of Xwhich is not C-normal. □
✓
✓
✓
Example 5.3. There exists an epi-compact space admitting a closed continuous image
which is not C-normal.
Proof. Take the spaces Xand Yas in Example 5.2. Now consider the function f:X→Y
given by:
f(x) = {id(x),if x∈2ω1×Σ2ω1;
0,otherwise.
Notice that fis continuous and f(X)=2ω1×Σ2ω1. Besides, if Fis closed in X, then
either f(F) = F∩(2ω1×Σ2ω1)or f(F) = (F∩(2ω1×Σ2ω1)) ∪ {0}. It follows that fis a
closed function. Finally, we know that Xis epi-compact and f(X)is not C-normal. □
✓
✓
✓
Any closed continuous function is quotient; from the previous result we conclude that C-
normality is not preserved under quotient functions. It happens that C-normality is also
not preserved under open perfect preimages. Indeed, take the proyection π: 2ω1×Σ2ω1→
Σ2ω1on the second factor. As 2ω1is compact, it follows from [5, Theorem 3.7.1.] that the
function πis perfect. However, the space Σ2ω1is C-normal while 2ω1×Σ2ω1=π−1(Σ2ω1)
is not C-normal.
Question 5.4. Suppose X×Kis C-normal for some compact K. Is it true that Xis
C-normal?
We will prove now that C-normality is not preserved under the union of two arbitrary
subspaces; we will use an example obtained in [4].
Example 5.5. There exists a non-C-normal space which is the union of a compact sub-
space and a locally compact subspace.
Proof. Consider the topological product (ω1+ 1) ×[0,1], the subspace R={ω1} × (0,1),
the space X=((ω1+ 1) ×[0,1])\R, and the space Y=X×(ω1+ 1). Then the
space Yis not C-normal (see [4]). Now, take A= (ω1+ 1) × {0,1} × (ω1+ 1) and
B= (ω1+ 1) ×(0,1) ×(ω1+ 1). Clearly Ais compact, Bis locally compact, and
Y=A∪B.□
✓
✓
✓
Question 5.6. Is there a non-C-normal Tychonoff space Xwhich is the union of two
C-normal closed subspaces?
Now we will answer in the positive the following question which is attributed to
Arhangel’skii in [7]; Is there a normal space which is not C-paracompact?
Example 5.7. There exists a normal space which is not C-paracompact.
[Revista Integración
Some topological properties of C-normality 101
Proof. Consider the normal space Σ2ω1. We claim that Σ2ω1is not C-paracompact.
Suppose that the space Σ2ω1is C-paracompact. Since 2ω1is C-compact, we can ap-
ply Proposition 3.7and the fact that the product of a compact space and a paracom-
pact space is paracompact (see [5, Theorem 5.1.36]), to conclude that 2ω1×Σ2ω1is
C-paracompact and thus C-normal; which we know is not true. Thus, the space Σ2ω1is
not C-paracompact. □
✓
✓
✓
Note that using the space described in examples 5.2 and 5.7 we can conclude that C-
paracompactness is not inherited by closed subspaces. It is worth to mention that Ex-
ample 5.7 also provides an epi-normal space which is not C-paracompact. This answers
another question from [7].
6. Epi-normal spaces
It follows from Examples 5.2, 5.3 and 5.5 that epi-normality is not necessarily preserved
under closed subspaces, unions, products, continuous and closed images, and inverse
images of perfect functions. Now we will analyze other properties of epi-normal spaces.
Proposition 6.1. Let Xbe an epi-normal space. If g:C→Ris a continuous function,
where C⊂Xis compact, then there exists a continuous function ˆg:X→Rsuch that
ˆg↾C=g.
Proof. Let f:X→Ybe bijective and continuous, for some normal space Y. Notice that
f↾C:C→f(C)is a homeomorphism. As Yis normal, the function h=g◦(f↾C)−1:
f(C)→Radmits a continuous extension ˆ
h:Y→R. We consider the continuous
function ˆg=ˆ
h◦f:X→R. Notice that
ˆg↾C= (ˆ
h◦f)↾C= (ˆ
h↾f(C))◦(f↾C) = g◦(f↾C)−1◦f↾C=g
is the required extension of g.□
✓
✓
✓
Corollary 6.2. If Xis epi-normal, then Xis Urysohn.
Proof. Given two distinct points x, y ∈X, by Proposition 6.1 we can take a continu-
ous function f:X→Rsuch that f(x)=2and f(y)=5. Then the open subsets
U=f−1((1,3)) and V=f−1((4,6)) of Xhave disjoint closures and contain xand y,
respectively. □
✓
✓
✓
The following example shows that, in general, epi-normal spaces are not necessary regular.
Example 6.3. Let X=Rand consider the sequence A={1/(n+ 1)}n∈N. Define a
topology in Xas the family of all sets of the form U\Bwhere B⊂Aand Uis open
in the usual topology of R. Clearly Xis epi-normal, because its topology is finer than
the usual topology. However, the space Xis not regular, because {0}and Acannot be
separated by disjoint open subsets.
Example 6.4. There exists a space Xwhich is neither C-normal nor Urysohn, but which
is the union of two epi-normal closed subspaces.
Vol. 38, No. 2, 2020]
102 I.E. Soberano-González, G. Delgadillo-Piñón & R. Rojas-Hernández
Proof. Let X= (R2\ {0})∪ {x1, x2}, where x1and x2do not belong to R2. Define a
topology in Xas follows. The space R2\ {0}endowed with its usual topology is an open
subspace of X. Besides, for each i= 1,2the point xihas as a basis of open neighborhoods
the family of all sets of the form
Un,i ={xi}∪{(x, y) : x2+y2<1/(n+ 1)2and (−1)iy > 0},
where n∈N. Note that x1and x2cannot be separated using neighborhoods with disjoint
closures, thus Xis not Urysohn. As an application of Corollary 6.2 we obtain that Xis
not epi-normal. Observe that Xis Fréchet-Urysohn, so we can apply Proposition 3.2 to
conclude that Xis not C-normal. Choose i∈ {1,2}. Let Ai={(x, y) : (−1)iy≥0}\{0}.
Note that Ai∪ {xi}admits a bijective continuous function onto the subspace Ai∪ {0}of
R2and hence is epi-normal. Therefore, the space X= (A1∪ {x1})∪(A2∪ {x2})is the
union of two closed epi-normal subspaces. □
✓
✓
✓
It happens that C-normal spaces are not necessarily Urysohn, as the following example
shows.
Example 6.5. There exists a C-normal space which is not Urysohn.
Proof. Consider the space ω1with the discrete topology. Let L=ω1+1 endowed with the
following topology. The space ω1is open in Land ω1has as a basis of open neighborhoods
the family of all sets (α, ω1], where α < ω1. Consider the open subspace O=L×ω1
of L×L. Let {A1, A2}be a partition of ω1into uncountable sets. Consider the space
X=O∪ {x1, x2}, where x1, x26∈ L×L, endowed with the following topology. The space
Ois open in Xand, for i∈ {1,2}the point xihas as a basis of open neighborhoods the
family of all sets of the form (U∩(Ai×ω1)) ∪ {xi}where Uis a neigborhood of (ω1, ω1)
in L×L. Note that Xis Hausdorff. Besides, the space Xis not Urysohn because the
points x1and x2cannot be separated by open sets in Xwith disjoint closures. However,
if we take Y=O⊕ {x1} ⊕ {x2}, then Yis normal and the restriction of the identity
function from Xonto Yto each compact subspace is a homeomorphism, that is, the
space Xis C-normal. □
✓
✓
✓
References
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[2] AlZahrani S. and Kalantan L., “Epinormality”, J. Nonlinear Sci. Appl., 9 (2016), No. 9,
5398-5402. doi: 10.22436/jnsa.009.09.08.
[3] Arkhangel’skii A.V., Topological Function Spaces, Kluwer Acad. Publ., Dordrecht, 1992.
[4] Buzkayova R.Z., “On the product of two normal spaces”, Moscow Univ. Math. Bull., 49
(1994), No. 5, 52-53.
[5] Engelking R., General Topology, Heldermann Verlag, Berlín, 1989.
[6] Saeed M.M., “Countable Normality”. arXiv:1709.10404.
[7] Saeed M.M., Kalantan L. and Alzumi H., “C-paracompactness and C2-paracompactness”,
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[Revista Integración
