

@
Appl. Gen. Topol. 23, no. 2 (2022), 325-331

doi:10.4995/agt.2022.17006

© AGT, UPV, 2022

Results about S2-paracompactness

Ohud Alghamdi a and Lutfi Kalantan b

a Department of Mathematics, Faculty of Science and Arts in Almandaq, Al-Baha Univesity,

P.O.Box 1988, Al-Baha 65581, Saudi Arabia (ofalghamdi@bu.edu.sa)
b Department of Mathematics, King Abdulaziz University, Saudi Arabia (lkalantan@kau.edu.sa

and lnkalantan@hotmail.com)

Communicated by M. A. Sánchez-Granero

Abstract

We present new results regarding S2-paracompactness, that we estab-
lished in [1], and its relation with other properties such as S-normality,
epinormality and L-paracompactness.

2020 MSC: 54C10; 54D20.

Keywords: separable; paracompact; S-paracompact; S2-paracompact; L-
paracompact; L2-paracompact; S-normal; L-normal.

1. Introduction

In this paper, we present some new results about S-paracompactness and
S2-paracompactness. First, we introduce the significant notations. An order
pair will be denoted by 〈x,y〉. The sets of positive integers, rational numbers,
irrational numbers and real numbers will be denoted by N,Q,P and R, respec-
tively. The closure and the interior of the subset A of a topological space X
will be denoted respectively by A and int(A). Throughout this paper, a T1
normal space is called T4 and a T1 completely regular space is called Tychonoff
space (T3 1

2
). In the definitions of compactness, countable compactness, para-

compactness, and local compactness we do not assume T2. Moreover, in the
definition of Lindelöfness we do not assume regularity. Also, the ordinal γ is
the set of all ordinal α such that α < γ. We denote the first infinite ordinal by
ω and the first uncountable ordinal by ω1.

Received 13 January 2022 – Accepted 30 August 2022

http://dx.doi.org/10.4995/agt.2022.17006


O. Alghamdi and L. Kalantan

Definition 1.1. A topological space X is called S-paracompact if there exist a
paracompact space Y and a bijective function f : X −→ Y such that for every
separable subspace A ⊆ X we have that f |A: A −→ f(A) is a homeomorphism.
Moreover, If Y is T2 paracompact, then X is S2-paracompact [1].

2. S2-paracompactness and other Topological Properties

2.1. S2-paracompactness and L2-paracompactness.

Recall from [5] that a topological space X is called L-paracompact if there
exist a paracompact space Y and a bijective function f : X −→ Y such that
f |B: B −→ f(B) is a homeomorphism for all Lindelöf subspace B of X. In
addition, if Y is T2 paracompact, then X is L2-paracompact.

Recall from [6] that a topological space X is called P-space if it is T1 and
every Gδ set is open. The countable complement topology defined on R, (R,CC)
(see [9, Example 20]), is an example of a space that is S2-paracompact but not
L2-paracompact. It is S2-paracompact because it is P-space, (see [1]), but not
L2-paracompact because it is Lindelöf and not paracompact space. In fact, it
is not even L-paracompact.
We still do not have an answer for the following question:
Does there exist an L-paracompact space which is not S-paracompact?

Theorem 2.1. If X is L-paracompact (resp. L2-paracompact) such that for
any separable subspace A ⊆ X there exists a Lindelöf subspace B ⊆ X such
that A ⊆ B, then X is S-paracompact (resp. S2-paracompact).

Proof. Let X be L-paracompact such that for any separable subspace A ⊆ X
there exists a Lindelöf subspace B ⊆ X such that A ⊆ B. Then, there exist
a paracompact space Y and a bijective function f : X −→ Y such that f |B:
B −→ f(B) is a homeomorphism for every Lindelöf subspace B ⊆ X. Let A
be any separable subspace of X. Then, there exists a Lindelöf subspace B of
X such that A ⊆ B. Then, f |A: A −→ f(A) is a homeomorphism. �

A similar proof as in Theorem 2.1 yields the following corollaries.

Corollary 2.2. If X is S-paracompact (resp. S2-paracompact) such that for
any Lindelöf subspace B ⊆ X, there exists a separable subspace A with B ⊆ A.
Then, X is L-paracompact (resp. L2-paracompact).

Recall from [7] that a space X is called C-paracompact if there exist a
paracompact space Y and a bijection f : X −→ Y such that f |K: K −→ f(K)
is a homeomorphism for every compact subspace K ⊆ X. Moreover, If Y is T2
paracompact, we say that X is C2-paracompact.

Corollary 2.3. If X is S-paracompact (resp. S2-paracompact) such that for
any compact subspace B ⊆ X, there exists a separable subspace A of X with
A ⊆ B. Then, X is C-paracompact (resp. C2-paracompact).

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 326


Results about S2-paracompactness

Application of Corollary 2.3:
Take (R,CC), the countable complement topology on R. Since A ⊂ R is com-
pact if and only if A is finite, we can say that every compact subspace is
contained in a separable subspace of R. Thus, (R,CC) is C2-paracompact.

Recall from [3, 4.4.F] that a space X is locally separable if each element of
X has a separable open neighborhood.

Theorem 2.4. Every S-paracompact (resp. S2-paracompact) hereditarly lo-
cally separable is L-paracompact (resp. L2-paracompact).

Proof. Let X be S-paracompact (resp. S2-paracompact) and hereditarly lo-
cally separable and let B be any Lindelöf subspace of X. Then, B is a locally
separable Lindelöf subspace of X. Pick Ux to be a separable open neighborhood
of each x ∈ B. Then, {Ux}x∈B is an open cover of B. Let U be a countable
open subcover of {Ux}x∈B and let Dx be a countable dense subset of each
Ux ∈ U. Then, D =

⋃
Dx is a countable dense subset of B, implying that B

is separable. Therefore, since every Lindelöf subspace of X is separable and
X is S-paracompact (resp. S2-paracompact), then X is L-paracompact (resp.
L2-paracompact). �

Problem 2.5. Does there exist a topological space that is L-paracompact (resp.
L2-paracompact) but not locally separable or not S-paracompact (resp. S2-
paracompact)?

Note that local separability is essential in Theorem 2.4. For example, (R,CC)
is S-paracompact not locally separable. Observe that (R,CC) is not L-paracompact.

Theorem 2.6. Let X be a topological space such that the only separable or
Lindelöf subspaces are the countable ones. Then, X is S-paracompact (resp.
S2-paracompact) if and only if X is L-paracompact (resp. L2-paracompact).

Proof. Let X be any topological space such that the only separable or Lindelöf
subspaces are the countable ones. Suppose that X is S-paracompact (resp. S2-
paracompact). If B is any Lindelöf subspace of X, then B is countable, imply-
ing that B is separable. Hence, X is L-paracompact (resp. L2-paracompact).
Conversely, suppose that X is L-paracompact (resp. L2-paracompact) and A
is any separable subspace of X. Then, A is countable, implying that A is
Lindelöf. Hence, X is S-paracompact (resp. S2-paracompact). �

Application of Theorem 2.6:
Consider ω1 with its usual ordered topology. Let A be any uncountable subset
of ω1. Then A is not bounded. Hence, {[0,α] : α < ω1} is an open cover of A
that has no countable subcover, which implies that A is not Lindelöf. Since ω1
satisfies the condition in Theorem 2.6, then ω1 is L2-paracompact because it is
S2-paracompact, (see [1]).

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 327


O. Alghamdi and L. Kalantan

A family {As}s∈S of subsets of a space X is called point-finite if for each
x ∈ X, the set {s ∈ S : x ∈ As} is finite, (see [3]).
Recall from [9] that a space X is metacompact if every open cover of X has a
point-finite open refinement.

Theorem 2.7. Any hereditarly metacompact L-paracompact (resp. L2-paracom-
pact) is S-paracompact (resp. S2-paracompact).

Proof. Let X be L-paracompact (resp. L2-paracompact) hereditarly metacom-
pact and let A be any separable subspace of X. Then, A is a separable meta-
compact subspace of X. Suppose that A is not Lindelöf. Then, there exists an
open cover of A, say W = {Wα : α ∈ Λ}, which has no countable subcover.
Let U be a point-finite open refinement of W. Then, U is uncountable by our
assumption. Let D be the countable dense subset of A. Hence, D∩U 6= ∅ for
all U ∈U implying that there exists d ∈ D contained in uncountable members
of U which contradicts the fact that U is a point-finite family. Hence, A is
Lindelöf, implying that X is S-paracompact (resp. S2-paracompact). �

Problem 2.8. Does there exist a topological space which is S-paracompact
(resp. S2-paracompact) but not hereditarly metacompact or L-paracompact
(resp. L2-paracompact)?

2.2. S2-paracompactness and Epinormality.

Definition 2.9. A topological space (X,τ) is epinormal if there exists a coarser
topology, say V, such that (X,V) is T4, (see [2]).

Since every epinormal space is Hausdorff as it is proved in [2], then the
countable complement topology on R, (R,CC), is an example of S2-paracompact
that is not epinormal. On the other hand, the following example shows that
there exists an epinormal space which is not S2-paracompact.

Example 2.10. Let A = {〈x, 0〉 : 0 < x ≤ 1} and B = {〈x, 1〉 : 0 ≤ x < 1}.

Figure 1. This figure illustrates the neighborhood system of
Strong Parallel Line Topology (X,σ).

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 328


Results about S2-paracompactness

The strong parallel line topology σ on X = A ∪ B is the unique topology
generated by the following neighborhood system:
For each 〈t, 0〉 ∈ A, let B(〈t, 0〉) = {U : U = {〈x, 0〉 : 0 ≤ a < x ≤ t}∪{〈x, 1〉 :
a < x < t}}, and for each 〈t, 1〉 ∈ B, let B(〈t, 1〉) = {V : V = {〈x, 1〉 : t ≤ x <
b ≤ 1}}, (see [9, Example 96]).
Since (X,σ) is separable and not paracompact space because it is a Hausdorff
and not regular topological space, then (X,σ) cannot be S2-paracompact.
Define τ on X to be the unique topology that is generated by the following
neighborhood system:
Every element in A has the same local base as σ and for each element 〈t, 1〉 ∈ B,
let B(〈t, 1〉) = {V : V = {〈x, 0〉 : t < x < b ≤ 1}∪{〈x, 1〉 : t ≤ x < b}}. The
topology (X,τ) is named weak parallel line, (see [9, Example 96]).

Figure 2. This figure illustrates the neighborhood system of
Weak Parallel Line Topology (X,τ).

Define a relation � on X as follows:
For 〈x,y〉 and 〈k,l〉 ∈ X, we write 〈x,y〉 � 〈k,l〉 if and only if either x < k
or x = k and y = 0 < l = 1, or x = k and y = l. Then, (X,τ) is a linearly
ordered topological space (LOTS). Since any LOTS is T4, we have (X,τ) is T4
and τ is coarser than σ, hence, we get that (X,σ) is epinormal.

Theorem 2.11. Any S2-paracompact Fréchet space is epinormal.

Proof. Let (X,τ) be S2-paracompact Fréchet space. Without loss of generality,
assume that (X,τ) is not normal. Let (Y, τ ′) be a T2 paracompact space and
let f : X −→ Y be a bijective function such that f |A: A −→ f(A) is a
homeomorphism for every separable subspace A ⊆ X. Then, f is continuous
since X is Fréchet. Consider V = {f−1(U) : U ∈ τ ′ }. Then, V is a topology
on X and since any open set in V is open in τ by continuity of f, we get that
V is coarser than τ. Observe that f : (X,V) −→ (Y, τ ′) is a homeomorphism.
Therefore, (X,V) is T2 paracompact and, hence, T4. �

For the converse of Theorem 2.11, we have the left ray topological space
defined on R, (R,L), as an example of epinormal Fréchet space that is not
S2-paracompat since it is separable and not paracompact space.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 329


O. Alghamdi and L. Kalantan

2.3. S2-paracompactness and S-normality.

Recall that a topological space X is S-normal if there exist a normal space
Y and a bijective function f : X −→ Y such that f |A: A −→ f(A) is a home-
omorphism for each separable subspace A of X, (see [4]). From the definition
of S-normality, it is clear that any S2-paracompact is S-normal. However, we
show in the following example that this relation is not reversible.

Example 2.12. An example of a T4 topological space that is S-normal but
not S2-paracompact is the sigma product Σ(0) as a subspace of 2

ω1 , where
2 = {0, 1} considered with the discrete topology. It is not S2-paracompact
since it cannot be condensed onto a T2 paracompact space, (see [8]).

Theorem 2.13. Let X be Fréchet and Lindelöf space such that any finite
subspace of X is discrete. X is S-normal if and only if X is S2-paracompact.

Proof. Let Y be a normal space and let f : X −→ Y be a bijective function
such that f |A: A −→ f(A) is a homeomorphism for each separable subspace
A of X. Without loss of generality, let X have more than one element. Thus,
Y is T1 since any finite subspace of X is separable and discrete. By continuity
of f and since X is Lindelöf, then Y is Lindelöf. Since Y is T3 Lindelöf, then
Y is T2 paracompact. Thus, X is S2-paracompact.
Conversely, assume that X is S2 paracompact. Let Y be a T2 paracompact
space and let f : X −→ Y be a bijective function such that f |A: A −→ f(A)
is a homeomorphism for each separable subspace A of X. Hence, since Y is T2
paracompact, then Y is T4. Therefore, X is S-normal. �

Recall that from [3] that a topological space X is locally metrizable if there
exists a metrizable open nighborhood for each x ∈ X..

Theorem 2.14. If X is hereditary Lindelöf, S2-paracompact and locally metriz-
able, then X is T2 paracompact and, hence, T4.

Proof. Set Ux to be a metrizable open nieghborhood of each x ∈ X. Since
X is Lindelöf, then there exists a countable set E such that X ⊆

⋂
x∈E Ux.

Now, since X is hereditary Lindelöf, then Ux is Lindelöf as a subspace of X for
every x ∈ X. Hence, Ux is separable being Lindelöf and metrizable for every
x ∈ X. Since X is S2-paracompact and separable, then X is T2 paracompact
and, hence, T4. �

Let (X,τ) be a topological space and let M be a proper nonempty subset
of X. The discrete extension of the topological space (X,τ) is defined by the
following neighborhood system:
For each x ∈ X \M, let B(x) = {{x}} and for each x ∈ M, let B(x) = {U ∈τ:
x ∈ U}. We denote the discrete extension of X by XM , (see [9]).

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 330


Results about S2-paracompactness

The following example shows that the discrete extension of S2-paracompact
need not to be S2-paracompact.

Example 2.15. Consider (R,RS), the rational sequence topology on R, (see
[9, Example 65]). Since RS is separable and not paracompact, then it is not
S2-paracompact. Also, because it is a Tychonoff locally compact space, we can
set X = R ∪{p} to be the one point compactification of it. X is T2 compact,
which implies that X is S2-paracompact. Consider XR, the discrete extension
of X. Since {p} is closed and open subset in XR, then R is a closed subspace
of XR. However, since (R,RS) is not normal, we conclude that XR cannot be
normal. Since XR is T2 and not normal space, then XR is not paracompact.
Since XR is separable as Q∪{p} is a countable dense subset of XR, then XR is
not S2-paracompact.

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© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 331