EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 2, 2021, 351-357
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Results on C2-paracompactness

Hala Alzumi1,2, Lutfi Kalantan1, Maha Mohammed Saeed1,∗

1 Department of Mathematics, King Abdulaziz University, P.O.Box 80203, Jeddah 21589,
Saudi Arabia
2 Department of Mathematics, Jeddah University, Jeddah, Saudi Arabia

Abstract. A C-paracompact is a topological space X associated with a paracompact space Y
and a bijective function f : X −→ Y satisfying that f �A: A −→ f(A) is a homeomorphism for
each compact subspace A ⊆ X. Furthermore, X is called C2-paracompact if Y is T2 paracompact.
In this article, we discuss the above concepts and answer the problem of Arhangel’skĭi. Moreover,
we prove that the sigma product Σ(0) can not be condensed onto a T2 paracompact space.

2020 Mathematics Subject Classifications: 54C10, 54D20

Key Words and Phrases: Normal, paracompact, sigma product, C-paracompact, C2-paracompact,
C-normal, sigma product, open invariant, Alexandroff duplicate

1. Introduction and preliminaries

In the present work, we give some new results about C-paracompactness and C2-
paracompactness [8] and answer a problem of Arhangel’skĭi. Also, we prove that the
sigma product Σ(0) can not be condensed onto a T2 paracompact space. Throughout this
paper, 〈x, y〉 denotes an ordered pair, N denotes the set of positive integers, Q denotes the
rational numbers, P denotes the irrational numbers, and R denotes the set of real numbers.
T2 denotes the Hausdorff property. A T4 space is a T1 normal space and a Tychonoff space
(T3 1

2
) is a T1 completely regular space. We do not assume Hausdorffness in the definition

of compactness, countable compactness, local compactness, and paracompactness. So, a
space is paracompact if any open cover has a locally finite open refinement. The regularity
of Lindelöfness’s definition is not assumed. The interior and the closure of a subset A of
a space X, are denoted by intA and A, respectively. An ordinal γ consists of all ordinal α
that satisfying α < γ. The first infinite ordinal is ω0, the first uncountable ordinal is ω1,
and the successor cardinal of ω1 is ω2.

We begin by recalling the following definition, see [8].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v14i2.3941

Email addresses: hala.100@hotmail.com (H. Alzumi),
lkalantan@kau.edu.sa (L. Kalantan), mmmohammed@kau.edu.sa (M. M. Saeed)

http://www.ejpam.com 351 c© 2021 EJPAM All rights reserved.



H. Alzumi, L. Kalantan and M. Mohammed Saeed / Eur. J. Pure Appl. Math, 14 (2) (2021), 351-357 352

Definition 1. (Arhangel’skĭi, 2016.)
A topological space X is called C-paracompact if there exist a paracompact space Y

and a bijective function f : X −→ Y such that the restriction f �A: A −→ f(A) is
a homeomorphism for each compact subspace A ⊆ X. A topological space X is called
C2-paracompact if there exist a Hausdorff paracompact space Y and a bijective function
f : X −→ Y such that the restriction f �A: A −→ f(A) is a homeomorphism for each
compact subspace A ⊆ X.

In [8, Theorem 2.2], the following theorem was proved.

Theorem 1. If X is Fréchet and C-paracompact (C2-paracompact), then any function
witnesses its C-paracompactness (C2-paracompactness) is continuous.

2. Results and Examples

Since the paracompactness is not multiplicative, it seems that both C-paracompactness
and C2-paracompactness are not multiplicative, but we still could not find a counterex-
ample. We introduce here a case where C-paracompactness and C2-paracompactness are
multiplicative.

Theorem 2. If X is C-paracompact (C2-paracompact) and Z is a compact T2 space, then
X × Z is C-paracompact (C2-paracompact).

Proof. Let Y be a paracompact (T2 paracompact) space and f : X −→ Y be a
bijective function such that the restriction f �A: A −→ f(A) is a homeomorphism for
each compact subspace A ⊆ X. Consider the product space Y × Z which is paracompact
(T2 paracompact), because the product of any paracompact space with a compact space
is paracompact, see [4, 5.1.36]. Define g : X × Z −→ Y × Z by g(〈x, i〉) = 〈f(x), i〉.
Then g is a bijective function and g = f × idZ , where idZ is the identity function on
Z. Let C be any compact subspace of X × Z. Then C ⊆ p1(C) × p2(C), where p1 and
p2 are the usual projection functions. p1(C) is a compact subspace of X and p2(C) is
a compact subspace of Z, thus p1(C) × p2(C) is a compact subspace of X × Z. Now,
f �p1(C): p1(C) −→ f(p1(C)) is a homeomorphism and idZ �p2(C): p2(C) −→ p2(C) is a
homeomorphism. Thus (f × idZ) �(p1(C)×p2(C)): p1(C)× p2(C) −→ f(p1(C))× p2(C) is a
homeomorphism. We conclude that g �C : C −→ g(C) is a homeomorphism because

g �C= ((f × idZ) �(p1(C)×p2(C))) �C .

Corollary 1. If X is C2-paracompact (C-paracompact), then so is X × I, where I is the
closed unit interval [0, 1] considered with its usual Euclidean metric topology.

We still do not know an answer of the converse of the above theorem which is the
following statement: If X × I is C2-paracompact, is then X C2-paracompact ? Observe
that if X is C2-paracompact and Y is T4, then the natural projection p : X × Y −→ Y



H. Alzumi, L. Kalantan and M. Mohammed Saeed / Eur. J. Pure Appl. Math, 14 (2) (2021), 351-357 353

may not be closed. For example, ω1 is C2-paracompact being T2 locally compact [8] and
ω1 + 1 is T2 compact, hence T4, but p : ω1 × (ω1 + 1) −→ ω1 + 1 is not closed, see [4,
3.10.16].

Referring to Theorem 1, we introduce here another case when a product of two C2-
paracompact spaces will be C2-paracompact.

Theorem 3. If X and Z are C2-paracompact spaces such that X is Fréchet and countably
compact, then X × Z is C2-paracompact.

Proof. Let Y and Y ′ be T2 paracompact spaces, f : X −→ Y and f ′ : Z −→ Y ′ be
bijective function such that the restriction of each of them on any compact subspace is a
homeomorphism. Define g : X×Z −→ Y ×Y ′ by g(〈x, z〉) = 〈f(x), f ′(z)〉, i.e., g = f ×f ′.
Then g is bijective. Now, X is Fréchet gives that f is continuous, see Theorem 1. Since X
is countably compact and f continuous surjective, then Y is countably compact. Hence Y
is compact because any T2 countably compact paracompact is compact [4, 5.1.20]. Since a
product of a T2 paracompact space with a T2 compact space is T2 paracompact [4, 5.1.36],
then Y ×Y ′ is T2 paracompact. Now, similar argument as in the proof of Theorem 2 shows
that the restriction of g on any compact subspace C of X × Z will be a homeomorphism.

Recall that a topological space (X , τ ) is called lower compact if there exists a coarser
topology τ ′ on X such that (X , τ ′ ) is T2-compact, [8]. It was proved in [8, 2.21] that “if
X is C2-paracompact countably compact Fréchet, then X is lower compact”. It turns out
that lower compactness is enough in Theorem 3.

Theorem 4. If X is lower compact and Z is C2-paracompact, then X×Z is C2-paracompact.

Proof. Let τ denote the topology on X. Let τ ′ be a T2 compact topology on X
coarser than τ . Pick a T2 paracompact space Y ′ and a bijective function f ′ : Z −→ Y ′

such that the restriction of f ′ on any compact subspace is a homeomorphism. Define
g : X×Z −→ X×Y ′ by g(〈x, z〉) = 〈x, f ′(z)〉, i.e., g = idX×f ′, where X in the codomain
is considered with the topology τ ′. Observe that the restriction of the identity function
idX : (X , τ ) −→ (X , τ ′ ) on any compact subspace A in (X , τ ) is a homeomorphism,
see [4, 3.1.13]. Also, the codomain of g, X × Y ′ is T2 paracompact being a product of
a T2 compact space (X , τ ′ ), with a T2 paracompact space Y ′, [4, 5.1.36]. Now, similar
argument as in the proof of Theorem 2 shows that the restriction of g on any compact
subspace C of X × Z will be a homeomorphism.

Here is an example showing that the Fréchet property is essential in Theorem 1.

Example 1. Consider ω2, the successor cardinal number of the cardinal number ω1. Let
[ω2]

≤ω1 = {E ⊂ ω2 : |E| ≤ ω1 }. Let i 6∈ ω2 and put X = {i} ∪ ω2. For each α ∈ ω2,
let {α} be open and an open neighborhood of i is of the form U = {i} ∪ (ω2 \ E) where



H. Alzumi, L. Kalantan and M. Mohammed Saeed / Eur. J. Pure Appl. Math, 14 (2) (2021), 351-357 354

E ∈ [ω2]
≤ω1. Then X is not Fréchet because i ∈ ω2 and the only convergent sequence is

the eventually constant sequence. (So, X is not even of countable tightness.)
Observe that X is T2 paracompact. BUT we will not treat X in this way. A subspace A

of X is compact if and only if A is finite. So, by [8, Theorem 2.7], X is C2-paracompact and
X = Y with the discrete topology and the identity function witness the C2-paracompactness
of X and clearly the identity function can not be continuous because X is not discrete.

Observe that X in Example 1 is not of countable tightness as i ∈ ω2 but there is no
countable subset A of ω2 satisfies i ∈ A. Here is an example showing that the countable
tightness property is not enough in Theorem 1.

Example 2. For each i ∈ N, let Xi = {ai} ∪ {ai,j : j ∈ N } be such that Xn ∩ Xm = ∅
for each n,m ∈ N with n 6= m. Let a 6∈ ∪i∈NXi and put X = {a} ∪ (∪i∈NXi). Generate
a topology on X by the following neighborhood system: For each i, j ∈ N, let B(ai,j) =
{{ai,j}}. For each i ∈ N, let B(ai) = {ai} ∪ {ai,j : j ≥ k, where k ∈ N }. For members of
B(a) we take all sets obtained from X by removing a finite numbers of Xi’s and a finite
number of points of ai,j in all the remaining Xi’s. So, if U ∈ B(a), then U is of the form
U = {a} ∪ (∪i∈(N\E)X

′
i) where E is a finite subset of N and X ′i = Xi \ Ei where Ei is a

finite subset of {ai,j : j ∈ N}. It is well-known that X is zero-dimensional normal space
which is not Fréchet [4, 1.6.19]. Let Z = X \ { ai : i ∈ N }. Then Z as a subspace of X
is not sequential [4, 1.6.20]. But since Z is countable, then it is of countable tightness.
Now, a subspace C of Z is compact if and only if C is finite. Since Z is also T1, then
by [8, Theorem 2.7], Y = Z with the discrete topology and the identity function witness
the C2-paracompactness of Z and since Z is not discrete, then the identity function is not
continuous.

Let X be any set containing more than one element. Fix an element p ∈ X. The
topology τ= {∅}∪ {W ⊆ X : p ∈W } is called the particular point topology on X, see [9].

Theorem 5. Let (X , τ ) be a Fréchet σ-compact non-compact space such that τ is
coarser than a particular point topology τ p on X, where p ∈ X, then (X , τ ) can not be
C-paracompact.

Proof. Suppose that (X , τ ) is C-paracompact. Pick a paracompact space Y and a
bijective function f : X −→ Y such that the restriction f �A: A −→ f(A) is a homeomor-
phism for each compact subspace A ⊆ X. Since X is Fréchet, then f is continuous, see
Theorem 1. So, for any non-empty open subset W of Y we have that f−1(W ) is open in
X, hence p ∈ f−1(W ) which gives that f(p) ∈W .

Now, write X =
⋃
n∈NXn where Xn is compact for each n ∈ N and Xn ⊂ Xn+1 for

each n ∈ N, i.e., the Xn’s are increasing. Since X is not compact, there exists an open
cover U = {Uα : α ∈ Λ } such that for any finite subset F of Λ there exists an element
x ∈ X such that x 6∈

⋃
α∈F Uα. Now, U is an open cover for X1 and X1 is compact. Let

F1 be a finite subset of Λ such that X1 ⊆
⋃
α∈F1

Uα = V1. Pick a2 ∈ X \ V1 and let i2 ∈ N
be the minimal so that a2 ∈ Xi2 , i.e., if j < i2, then a2 6∈ Xj . Now, U is an open cover for



H. Alzumi, L. Kalantan and M. Mohammed Saeed / Eur. J. Pure Appl. Math, 14 (2) (2021), 351-357 355

Xi2 and Xi2 is compact. Let F2 be a finite subset of Λ such that Xi2 ⊆
⋃
α∈F2

Uα = V2.
If m ∈ N so that am ∈ X, im ∈ N, Fm finite subset of Λ, and Vm are chosen, then Pick
am+1 ∈ X \ Vm and let im+1 ∈ N be the minimal so that am+1 ∈ Xim+1 . Continue,
U is an open cover for Xim+1 and Xim+1 is compact. Let Fm+1 be a finite subset of Λ
such that Xim+1 ⊆

⋃
α∈Fm+1

Uα = Vm+1. So, we have constructed two countably infinite
families of Xim ’s, Vm’s such that Xim ⊂ Xim+1 and Xim+1 \ Vm 6= ∅ for each m ∈ N as
am+1 ∈ Xim+1 \ Vm.

Now, for each m ∈ N, f �Xim
: Xim −→ f(Xim) is a homeomorphism, where i1 = 1. We

have Vm∩Xim+1 is open in Xim+1 for each m ∈ N, thus f(Vm∩Xim+1) is open in f(Xim+1)
for each m ∈ N. Hence, for each m ∈ N there exists an open subset Wm of Y such that
Wm∩ f(Xim+1) = f(Vm∩Xim+1). Observe that f(am+1) 6∈Wm for each m ∈ N. Since the
family {Vm : m ∈ N } is an open cover for X, then we have that the family {Wm : m ∈ N }
is an open cover for Y consisting of distinct proper subsets of Y . Since each non-empty
open subset of Y must contain the element f(p), then the open cover {Wm : m ∈ N } of
Y has no locally finite open refinement, which is a contradiction. Therefore, (X , τ ) is
not C-paracompact.

Non-compactness assumption is essential in Theorem 5, for example, consider on R
the topology τ= {∅,R, {p}}, where p ∈ R.

The following example answers three kinds of invariants. We used two well-known
spaces, the Alexandroff duplicate space and the closed extension space. Recall that for
any T1 space X, let X ′ = X × {1}. Let A(X) = X ∪X ′. For simplicity, for an element
x ∈ X, we denote the element 〈x, 1〉 in X ′ by x′ and for a subset B ⊆ X let B′ = {x′ :
x ∈ B} = B × {1} ⊆ X ′. For each x′ ∈ X ′, let B(x′) = {{x′}}. For each x ∈ X, let
B(x) = {U ∪ (U ′ \ {x′}) : U is open in X with x ∈ U }. Let τ denote the unique topology
on A(X) which has {B(x) : x ∈ X} ∪ {B(x′) : x′ ∈ X ′} as its neighborhood system. A(X)
with this topology is called the Alexandroff Duplicate of X [3]. In [8], it was shown that
“ if X is C2-paracompact, then so is its Alexandroff duplicate A(X).”.

Example 3. Consider the Alexandroff duplicate space A(R) of R with its usual metric
topology. It is C2-paracompact, see [8, Theorem 28]. Now, let i =

√
−1 6∈ R and put

X = R∪{i}. Let τ be the closed extension topology on X generated from R with its usual
metric topology and i. So, τ= {∅} ∪ {W ∪ {i} : W ⊆ R;W is open in the usual metric
topology }.
(X , τ ) is not C-paracompact because it is Fréchet, being first countable, non-compact,
and coarser than the particular point topology on X where the particular point is i, see
Theorem 5. Define g : A(R) −→ X by

g(x) =

{
i ; if x ∈ R′
x ; if x ∈ R

g is an open surjection function. Thus C-paracompactness and C2-paracompactness are
neither invariant, open invariant, nor quotient invariant.



REFERENCES 356

Now we show that C-paracompactness and C2-paracompactness are both not hered-
itary. Recall that a space X is called C-normal if there exist a normal space Y and a
bijective function f : X −→ Y such that the restriction f �A: A −→ f(A) is a homeomor-
phism for each compact subspace A ⊆ X [1]. It is clear that any C2-paracompact space is
C-normal [8].

Example 4. Consider 2ω1, where 2 = {0, 1} with the discrete topology. Consider the
subspace of 2ω1 consisting of all points with at most countably many non-zero coordinates,
i.e., the sigma product Σ(0). Put X = 2ω1 × Σ(0). Raushan Buzyakova proved that X
can not be mapped onto a normal space Y by a bijective continuous function [2]. In [7],
M. Saeed proved that X is not C-normal, hence X is not C2-paracompact. Since X is a
Tychonoff non-compact space, any compactification of X is C2-paracompact while X is
not.

We still do not know if C-paracompactness (C2-paracompactness) is hereditary with
respect to closed subspaces or not.

Now, here is our first main result. Arhangel’skĭi stated the following problem, see
[8]: “Is there a T4 space which is not C2-paracompact?”. We will answer this problem in
positive.

Example 5. Consider the sigma product Σ(0) as a subspace of 2ω1, where 2 = {0, 1} with
the discrete topology, see Example 4. We have that Σ(0) is T4 [5, Theorem 7.4], countably
compact [6, Theorem 6.10], Fréchet [4, 3.10.D], hence it is a k-space [4, 3.10.D]. Also
Σ(0) is not paracompact because it is contained a copy of ω1 as a closed subspace [5,
Theorem 7.2]. Suppose that Σ(0) is C2-paracompact. By Theorem 2, X = 2ω1 × Σ(0) is
C2-paracompact. This contradicts M. Saeed’s result [7] and Buzyakova’s result [2] because
any T2 paracompact space is normal.

Here is our second main result. Recall that a function f : X −→ Y is called conden-
sation if it is bijective and continuous. The sigma product Σ(0) is a k-space [4, 3.10.D].
Considering the theorem “a function f of a k-space X to a topological space Y is con-
tinuous if and only if for every compact space C ⊆ X the restriction f �C : C −→ Y is
continuous”, [4, 3.3.21], we conclude the following:

Corollary 2. The sigma product Σ(0) can not be condensed onto any T2 paracompact
space.

Open Problem: Is C2-paracompactness multiplicative ?

References

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[6] W W Comfort Kunen K. Handbook of Set Theoretic Topology. Amsterdam, 1984.

[7] Maha Saeed. Countable normality, 2018.

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[9] L Steen and J A Seebach. Counterexamples in Topology. Dover Publications, 1995.