Int. J. Anal. Appl. (2022), 20:66

Mappings and Finite Product of Pairwise Expandable Spaces

Jamal Oudetallah1, Iqbal M. Batiha2,3,∗

1Department of Mathematics, Faculty of Science and Information Technology, Irbid National

University, P.O. Box 2600, Irbid, P.C. 21110, Jordan
2Department of Mathematics, Faculty of Science and Information Technology, Al-Zaytoonah

University of Jordan, P.O. Box 130 Amman 11733, Jordan
3Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 346, UAE

∗Corresponding author: i.batiha@zuj.edu.jo

Abstract. In this work, we intend to study certain mappings in bitopological spaces such as pairwise

perfect and pairwise countably perfect mappings that possess the property (E). Some properties of

such mappings are provided which have helped us to obtain some finite product theorems concerning

with pairwise expandable and almost pairwise expandable spaces. Two illustrative examples are given

to demonstrate the effectiveness of some proposed results.

1. Introduction

A bitopological space (X,τ1,τ2) is a non-empty set X with two arbitrary topologies τ1,τ2. The

idea of the bitopological space was induced by topologies generalized by the following two sets:

Bρ� = {y ∈ X | ρ(x,y) ≤ �}

and

Bδ� = {y ∈ X | δ(x,y) ≤ �},

where ρ and δ are quasi-metric spaces of X with ρ(x,y) = δ(y,x). From 1963, when Kellay introduced

the concept of bitopological space, several topological properties, which are already included in a single

topology, are generalized into bitopological spaces [1]. Some of these properties are compactness,

Received: Nov. 3, 2022.

2010 Mathematics Subject Classification. 54A05.
Key words and phrases. pairwise expandable; pairwise paracompact; pairwise subparacompact; pairwise metacompact;

bitopological space.

https://doi.org/10.28924/2291-8639-20-2022-66
ISSN: 2291-8639

© 2022 the author(s).

https://doi.org/10.28924/2291-8639-20-2022-66


2 Int. J. Anal. Appl. (2022), 20:66

paracompactness, separation axioms, connectedness, some special types of functions and many others

[1–4]. Many authors studied and investigated these bitopological spaces after Kelly, like Fletcher et

al. [5], Birsan [6], Reilly [7], Datta [8], Hdeib and Fora [9], Bose et al. [10], Killiman and Salleh [11],

Abushaheen et al. [12], and Qoqazeh et al. [13]. For further exploration in this field, this work aims

to present some properties of the pairwise perfect and the pairwise countably perfect mappings for

the purpose of using them to obtain some novel finite product theorems concerning with pairwise

expandable and almost pairwise expandable spaces.

To go forward in this paper and for more simplification, we state some notations and notions which

will be used later on. In particular, the closure and interior of the set A will be denoted respectively

by CL(A) and Int(A) with noting that (X,τ) is a topological space. Besides, if A is a subset of a

bitopological space X = (X,τ1,τ2), then the relative topology, which is a subspace topology of X, on

the set A inherited by τ will be denoted by τA. The cardinality of the set ∆ will be denoted by |∆|.
The sets R, Q, N, and Z will denote the sets of real numbers, rational numbers, natural numbers and
integer numbers, respectively. On the other hand, ω0 and ω1 will denote respectively the first two

uncountable ordinals, and m will denote generally for an infinite cardinal. Finally, the terms τu, τdis,

τcof and τcoc will denote the usual, discrete, cofinite and the co-countable topologies, respectively.

2. Basic Definitions

In the following content, we state certain definitions and preliminaries associated with the bitopo-

logical space for completeness.

Definition 2.1. [14] A collection subset F̃ = {Fα : α ∈ ∆} of a bitopological space (X,τ1,τ2) is said
to be pairwise locally finite if for each x ∈ X there exist an τ1–open set U containing x such that U
intersects only finitely many members of F̃, or there exist τ2– open set V containing x such that V

intersects only finitely many members of F̃.

Definition 2.2. [15] A P–open cover Ṽ of a bitopological space (X,τ1,τ2) is called parallel refinement

of a P–open cover Ũ of X if each τi–open set of Ṽ is contained in some τi–open set of Ũ, where

i = 1, 2.

Definition 2.3. [8] A bitopological space X is called P–m–paracompact, if every P–open cover Ũ of

X, so that
∣∣Ũ∣∣ ≤ m, has a pairwise locally finite open parallel refinement. If m = ω0, then the space

X is called P–countably paracompact. If the space X is P–m–paracompact for every m, then X is

called P–paracompact.

Definition 2.4. [14] Let m be an infinite cardinal, then the bitopological space (X,τ1,τ2) is called

τι–m–expandable space with respect to τj if for every τι–locally finite F̃ = {Fα : α ∈ ∆} with |4|≤ m,
there exist τj–locally finite collection G̃ = {Gα : α ∈ ∆} of open subsets of X such that Fα ⊂ Gα for
all α ∈ ∆ and for i 6= j, where i, j = 1, 2.



Int. J. Anal. Appl. (2022), 20:66 3

Definition 2.5. [14] A bitopological space (X,τ1,τ2) is called τi–expandable with respect to τj, if it

is an τi–m–expandable for every cardinal m and i 6= j where i, j = 1, 2.

Definition 2.6. [14] A bitopological space (X,τ
1,τ2) is called a pairwise expandable (or simply P–

expandable), if it is P–T2–space and it is τ1–expandable with respect to τ2 and τ2– expandable with

respect to τ1.

Definition 2.7. Let X = (X,τ1,τ2), Y = (Y,σ1,σ2) be two bitopological spaces and f : X −→ Y be
a given map. Then f is called pairwise closed (P–closed) map if it maps a τi–closed subset of X onto

a σi–closed subset of Y , for each i = 1, 2.

Definition 2.8. let f : X −→ Y be a P–closed, P–continuous map from a bitopological space
X = (X,τ1,τ2) onto a bitopological space Y = (Y,σ1,σ2) such that f−1(y) is compact for each

y ∈ Y . Then the map f is called pairwise perfect (P–perfect) map.

Definition 2.9. According to definition 2.2 above, if the preimage f−1(Y ) is P–countably compact

for each y ∈ Y then f is called a pairwise countably perfect (P–countably perfect) map.

Definition 2.10. The topologies τi and τj on a nonempty set X are said to have the property (E) if

A ∈ λC(X,τi ) and B ∈ λC(X,τj) imply A∩B ∈ λC(X,τi ).

3. Main Theoretical Results

In this part, we intend to state and derive some novel significant theoretical results in light of the

aforesaid definitions. At the beginning, one can observe that it is easy to prove the following two

lemmas:

Lemma 3.1. Let f : (X,τ1,τ2) −→ (Y,σ1,σ2) be a P–countably perfect map. Then we have:

• If F̃ = {Fα : α ∈ ∆} be a P–locally finite collection of subsets of Y , then f−1(F̃ ) = {f−1(Fα) :
α ∈ ∆} is a P–locally finite collection of X.
• If F̃ = {Fα : α ∈ ∆} be a P–locally finite collection of subsets of X, then f (F̃ ) = {f (Fα) :
α ∈ ∆} is also P–locally finite collection of subsets of Y .

Lemma 3.2. Let f : (X,τ1,τ2) −→ (Y,σ1,σ2) be a P–closed map, if G̃ = {Gα : α ∈ ∆} be a
P–locally finite of subsets of X and Ṽ = {Vα : α ∈ ∆} be a σi–open in Y for i = 1, 2 such that
Vα ⊆ f (Gα) for α ∈ ∆, then the family {f (Gα) : α ∈ ∆} is P–locally finite collection of subsets of Y .

Next, based on the two lemmas provided above, we state and prove the following important theorem

that concerns with the P–m-expandable space.

Theorem 3.1. Let f be a P–countably perfect map from a bitopological space X = (X,τ1,τ2) onto a

bitopological space Y = (Y,σ1,σ2). Then X is P–m–expandable if and only if Y is P–m–expandable.



4 Int. J. Anal. Appl. (2022), 20:66

Proof. ⇒) Suppose X = (X,τ1,τ2) is P–m–expandable, and F̃ = {Fα : α ∈ ∆} is a P–locally finite
collection of subsets of Y with |∆| ≤ m. Then, by Lemma 3.2, the following mapping:

f−1(F̃ ) = {f−1(Fα) : α ∈ ∆}

will be a P–locally finite collection of subsets of X with |∆| ≤ m. Now, since X is P–m–expandable,
then there exists a P–locally finite collection G̃ = {Gα : α ∈ ∆} of open subsets of X such that
f−1(Fα) ⊆ Gα, for each α ∈ ∆. Set

Vα = Y − f (X −Gα), α ∈ ∆.

Consequently, we have the following claim:

Claim Fα ⊆ Gα.
To prove this claim, we have clearly:

X −Gα ⊆ X − f−1(Fα),

and hence f (X −Gα) ⊆ Y −Fα. Again, we have:

Fα ⊆ Y − f (X −Gα).

This consequently implies that Fα ⊆ Vα, for each α ∈ ∆.
Now, it remains to show that Ṽ = {Vα : α ∈ ∆} is a P–locally finite collection of open subsets of Y .
For this purpose, we should note that Vα is σi–open in Y , for i = 1, 2. Now, since f is P–closed map,

Vα ⊆ f (Gα), and by Lemma 3.2, the family {f (Gα) : α ∈ ∆} will be a P–locally finite collection of
subsets of Y . Therefore, we conclude that Ṽ is locally finite collection of subsets of Y , which implies

that Y is indeed a P–m–expandable.

⇐) In order to show the converse inclusion, we assume that Y is P–m–expandable and F̃ = {Fα :
α ∈ ∆} is a P–locally finite collection of subsets of X such that |∆| ≤ m. By Lemma 3.1, we can
deduce that:

f (F̃ ) = {f (Fα) : α ∈ ∆}

represents a P–locally finite collection of subsets of Y . Hence, there exists P–locally finite collection

of subsets of σi–open subsets of Y , for i = 1, 2, say G̃ = {Gα : α ∈ ∆}, such that Fα ⊆ Gα for each
α ∈ ∆. Then:

Fα ⊆ f−1 (f (Fα)) ⊆ f−1(Gα).

Since f is P–continuous, we get f−1(Gα) is τi–open in X for i = 1, 2. Now, by Lemma 3.1, the

following collection

f−1(G̃) = {f−1(Gα) : α ∈ ∆}

will be a τi–open locally finite collection for i = 1, 2. Hence, X is P–m–expandable. �

In the same context, we should notice that the P–closed continuous image of the P–expandable

space need not be P–expandable. However, The following example aims to illustrate this assertion.



Int. J. Anal. Appl. (2022), 20:66 5

Example 3.1. Let W = (W,τ,τ) where W −[0,ω1]×[0,ω1) and τ be an order topology define on W
and X = W ×N where N = (N,τdis,τdis) and τdis is a discrete topology. Clearly, W is P–countably
compact and hence X is p–expandable. Now, let

f1 : (X,τ ×τdis) −→ (Y,σ1)

be a P–closed onto continuous map. We have illustrated that Y is not ωo–expandable. Hence,

f2 : (X,τ ×τdis,τ ×τdis) −→ (Y,σ1,σ2)

is a P–closed onto continuous map and then Y is not P–ωo–expandable.

On the other hand, in order to turn into the other theoretical results, we will recall bellow the

following definition that relates with the P–expandable space.

Definition 3.1. Let X = (X,τ1,τ2) be a bitopological space. A τ1–subset F is called P–expandable

relative to X if and only if every P–open, P–Aσ–cover of F in X has a P–open, P–locally finite

refinement in X.

Thus, we are now ready to introduce the following important results that deal with the P–expandable

space.

Lemma 3.3. Let M be a τi–closed expandable subset of a bitopological space X = (X,τ1,τ2). If F

is τi–closed in Int(M), then F is P–expandable relative to X.

Proof. Let Ũ be a P–open, P–Aσ–cover of F in X. Then:

B̃ = {U ∩M : U ∈ Ũ}∪ (M −F )

will be an P–open, P–Aσ–cover of M. Since M is τi–expandable, then there exists a P–open, P–

locally finite refinement of B̃, say Ṽ , such that for each V ∈ Ṽ , there exists B ∈ B̃ such that V ⊆ B.
Now, let

W̃ = {V ∩ Int(M) : v ∈ Ṽ},

then W̃ will be a P–open locally finite refinement of Ũ in X. Therefore, F is indeed a P–expandable

relative to X. �

Theorem 3.2. Let i = 1, 2 and f : (X,τ1,τ2) −→ (Y,σ1,σ2) be a P–continuous map from a
bitopological space X onto a bitopological space Y . Then f is P–closed if and only if for each y ∈ Y
and each τi–open set U in X such that f−1(y) ⊆ U, there exists a σi–open set Oy in Y containing y
and f−1(Oy ) ⊆ U.

Proof. ⇒) Let i = 1, 2 and suppose f is P–closed. Let y ∈ Y and let U be any τi–open set such that
f−1(y) ⊆ U. If one lets A = X −U, then f (A) = f (X −U) will be σi–closed in Y . Moreover, letting

Oy = Y − f (X −U)



6 Int. J. Anal. Appl. (2022), 20:66

yields Oy to be σi–open in Y and y ∈ Oy, which consequently implies that f−1(Oy ) ⊆ U. Hence, if
we let t ∈ f−1(Oy ), then f (t) ∈ Oy and f (t) /∈ f (X −U), which implies t /∈ X −U, and therefore
t ∈ U.
⇐) Conversely, let A be any τi–closed subset of X and y ∈ Y − f (A). Since f is onto, then there
exists x ∈ X such that y = f (x). Now, we have the following claim:
Claim f−1(y) ∩A = φ.
To prove this claim, suppose that f−1(y) ∩A 6= φ. Then, there exists z ∈ f−1(y) and z ∈ A, which
implies that f (z) = y. But f (z) = Y − f (A), which gives that f (z) /∈ f (A). Hence, z /∈ A. This
yields a contradiction.

Thus, f−1(y)∩A = φ or f−1(y) ⊆ X−A. Consequently, due to X−A is τi–open, then by assumption
there exists a σi–open set Oy in Y such that f−1(Oy ) ⊆ X −A. Now, we have:

y ∈ Oy ⊆ f (X −A) = Y − f (A).

This implies that Y − f (A) is a σi–open subset of Y , which means that f (A) is σi–closed. Hence, we
have f is P–closed. �

Corollary 3.1. Let X be a P–compact space and Y = (Y,σ1,σ2) be any bitopological space. Then,

the projection map π : X ×Y −→ Y is P–closed map.

Proof. The proof follows directly from Theorem 3.2. �

Theorem 3.3. Let X = (X,τ1,τ2) be a P–expandable space and Y = (Y,σ1,σ2) be a P–compact

space. Then, X ×Y is P–expandable.

Proof. The projection map π : X ×Y −→ X is a P–closed map by Corollary 3.1 and because

π−1(x) = {x}×Y ∼= Y

is P–compact, for each x ∈ X. Therefore, Y is P–countably compact, which implies that π is a
P–countably perfect map. Hence, by Theorem 3.2, we conclude that X ×Y is P–expandable. �

It is worth mentioning that if X and Y are two P–expandable bitopological spaces, then X×Y need
not be in general P–expandable. However, this assertion can be illustrated by the following example.

Example 3.2. Suppose X = (X,τs,τs) so that τs represents the Sorgenfrey topology. Due to X is

P–Lindelof and P–T3–space, then X is P–paracompact and thus P− expandable. But X ×X is not
P–expandable since the collection F̃ = {(x,−x) : x ∈ X} is P–locally finite. However, one can show,
by a category argument, that there is no P–locally finite collection of open sets {Gx : x ∈ X} such
that (x,−x) ∈ Gx, for each x ∈ X.

Theorem 3.4. Let X = (X,τs,τs) be a P–regular, P–subparacompact and P–expandable bitopolog-

ical space and Y = (Y,σ1,σ2) be a P–expandable bitopological space. Then X ×Y is P–expandable



Int. J. Anal. Appl. (2022), 20:66 7

if and only if for each x ∈ X there is a τi–open set U(x) of X such that CL(U(x) is P–expandable,
for each i = 1, 2.

Proof. ⇒) Suppose X ×Y is P–expandable. Let x ∈ X and take X to be the τi–open set containing
x. This implies that CL(X) ×Y = X ×Y is P–expandable.
⇐) Conversely, suppose that for each x ∈ X there exist a τi–open set of x, U(x), such that CL(U(x))×
Y is P–expandable. It suffices to show that the following projection map:

π : X ×Y −→ X

has property (E). Let Ũ be a P–open, P–Aσ–cover of X×Y . Now, by P–regularity, and due to there
is a τi–open set of x, U(x), such that x ∈ U(x), for each x ∈ X and for i = 1, 2, then there is a
τi–open set V (x) of x such that:

x ∈ V (x) ⊆ CL(V (x)) ⊆ U(x) ⊆ CL(U(x)).

Then

V (x) ×Y ⊆ CL(V (x)) ×Y ⊆ CL(U(x)) ×Y.

By Lemma 3.3, CL(V (x)) ×Y is P–expandable relative to X×Y , which consequently implies that Ũ
has a P–open, P–locally finite refinement in X ×Y , say Ã(x) such that:

CL(V (x)) ×Y ⊆∪{Ã(x) : x ∈ X}.

Now, we have:

à = ∪{Ã(x) : x ∈ X}

is P–open, P–refinement of Ũ. Further, Ṽ = {V (x) : x ∈ X} is a P–open cover of X such that:

π−1(V (x)) = V (x) ×Y ⊆ ∪
x∈X

Ã(x),

where Ã(x) is a P–locally finite subfamily of Ã. Since X is P–subparacompact, then Ṽ is a P–open,

P−Aσ–cover of X. Therefore, we have for each P–open, P–Aσ–cover Ũ of X×Y , there is a P–open,
P–Aσ–cover Ṽ of X and P–open refinement à of Ũ such that π−1(V (x)) is contained in the union of

P–locally finite subfamily of Ã, for each x ∈ X and V (x) ∈ Ṽ . Hence, π has property (E), and X×Y
is P–expandable space. �

Corollary 3.2. Let X = (X,τs,τs) be a P–paracompact, P–locally compact bitopological space and

Y = (Y,σ1,σ2) be a P–expandable bitopological space. Then X ×Y is P–expandable.

Proof. The proof follows immediately from Theorem 3.3. �



8 Int. J. Anal. Appl. (2022), 20:66

4. Conclusions

This paper has studied and explored some certain mappings in bitopological spaces like the pairwise

perfect and pairwise countably perfect mappings. In order to gain some finite product theorems con-

cerning with pairwise expandable and almost pairwise expandable spaces, some significant properties

of such mappings have been stated and derived well. Some illustrative examples have been provided

for completeness.

Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi-

cation of this paper.

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https://doi.org/10.28919/jmcs/6294

	1. Introduction
	2. Basic Definitions
	3. Main Theoretical Results
	4. Conclusions
	References