EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 2574-2585
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A New Outlook on Omega Closed Functions in
Bitopological Spaces and Related Aspects

Ali A. Atoom1,∗, Hamza Qoqazeh2, Maryam M Alholi3, Eman ALmuhur4,
Eman Hussein5, Anas A. Owledat6 , Abeer A. Al-Nana7

1 Mathematics, Science, Ajloun National University, Ajloun, Jordan
2 Mathematics, Science and Information Technology, Irbid National University, Irbid,
Jordan
3 Applied, Taibah University, Al Ula, Saudi Arabia
4 Mathematics, Arts and Science, Applied Science Private University, Amman, Jordan
5 Mathematics, Arts and Science, Amman Arab University, Amman, Jordan
6 Ministry of Education, Amman, Jordan
7 Mathematics, Science, Prince Sattam Bin Abdulaziz University, Alkharj, Saudi Arabia

Abstract. Many studies have employed a variety of techniques to further investigate topological
space, particularly the notion of bitopological spaces, due to the significance of topological space
in data processing as well as certain implementations. Numerous extended topological structures
have been laid out subsequently. Of those abstractions, functions in topology was one of which
was most noteworthy. In order to assist in this trend, we focused our research on the idea of
open and closed sets, which is one of the strongest techniques available to present scientists for
the study of computer graphics and digital topology.New functions, pairwise ω−closed functions,
which are strictly weaker than pairwise closed functions, will be introduced in this study. By
applying the P̋−space definition, whose is a P−space modification. Additionally, we establish
different projection and product theories pertaining to pairwise Lindel

..
of and pairwise paracompact

spaces utilizing P̋−spaces. We analyze images and inverse images that have been chosen topological
attributes for every one of these functions. In the final analysis, we explore several counterexamples
that correspond to the offered definitions and theorems.

2020 Mathematics Subject Classifications: 54E55,54B10,54D30

Key Words and Phrases: Bitopological spaces, P̋−space, pair−Lindel
..
of, pair−ω−closed func-

tions, perfect function, pair−ω−continuous functions, pair- paracompact

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5347

Email addresses: aliatoom@anu.edu.jo (A. Atoom),
hhaaqq983@gmail.com (H. Qoqazeh), Mholi@taibahu.edu.sa (M. Alholi),
e almuhur@asu.edu.jo (E. ALmuhur), e.hussein@aau.edu.jo (E. Hussein),
emanbasssam@gmail.com (A. Owledat), a.alnana@psau.edu.sa (A. Al-Nana)

https://www.ejpam.com 2574 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



A. Atoom et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2574-2585 2575

1. Introduction

Several broad topological configurations have been explored subsequently. In light
of the topological space’s significance in analysis and various other uses, see [2, 3, 5].
One of the most fundamental topological space improvements is represented by the closed
functions. General topology informs us that closed sets are crucial for the creation of
new set forms and have vital topological traits. To expand on multiple features of closed
functions, ω−closed functions are primarily included in the topology. Compactness and
Lindel

..
of are the fundamental elements in standardized topology. Additionally, topology

and closed function theories have a frequent application in mathematical evaluation and
logical arithmetic correspondingly. Both of these notions are also very useful in real-world
applications. A novel kind of mappings known as ω−closed mappings, which are precisely
weaker than closed mappings, was created by [11] in 1982. He then goes over a few more
situations that have relevance to the definitions and theorems which are presented, as he
proposed the subsequent definitions of ω−open and ω−closed sets. If J has all of its
condensation points, then it is commonly referred to as being ω−closed. ω−open is the
complement of an ω−closed set. likewise the intersection of all ω−closed sets that con-
tain J will be indicated by clω J . The idea of the existence of a bitopological space was
initially introduced by [13] in 1963. Since then, other single topological qualities, includ-
ing Lindel

..
ofness, mapping types, separation axioms, compactness and metacompactness,

have also been stretched to bitopological spaces.We are going to utilize pairwise Lindel
..
of

as pair-Lindel
..
of during the course of this investigation, and pair- signifies pairwise. The

fundamental definitions employed in this investigation are presented in Section 2. We
demonstrate some properties of pair−ω−closed functions in Section 3. The association
between particular weakened versions of pairwise closed functions and pair−ω−closed
functions is illustrated with several instances in. Subsequently, the more complex charac-
teristics of the pair−ω−closed functions, notably their product and projection, are covered
in Section 4. In the end, in Section 5, we go through a variety of counterexamples that
are pertinent to the definitions and theorems offered in the earlier sections.

2. Basic definitions and preliminary remarks

Some key ideas and details that were employed in the research are presented in this
part.

Definition 1. [4] A function Υ : (D,κ1 , κ2) → (G, υ1 , υ2) is referred to as pair−continuous,

whether Υ1 : (D,κ1) → (G, υ1) and Υ2 : (D,κ2) → (G, υ2) are continuous functions.

Definition 2. [8] A function Υ : (D,κ1 , κ2) → (G, υ1 , υ2) is referred to as pair−closed,

if Υ1 : (D,κ1) → (G, υ1) and Υ2 : (D,κ2) → (G, υ2) are closed functions.
That is cruel H1is closed in κ1 ,then Υ(H1) is closed in υ1 , and if H2 is closed in κ2 ,

then Υ(H2) is closed in υ2 .



A. Atoom et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2574-2585 2576

Definition 3. [15] A cover T of the bitopological space (D,κ1 , κ2) has been referred to
κ1κ2− open if T ⊂ κ1 ∪ κ2. Additionally, T contains at least one−nonempty member of
κ2 ,it is regarded as pair−open.

Definition 4. [12] If any pair−open cover of a bitopological space has a countable subcover,
the space is commonly referred to as a pair−Lindel

..
of.

Definition 5. [13] If any κ1κ2−open cover of a bitopological space has a countable sub-
cover, the space is commonly referred to as a s−Lindel

..
of.

Definition 6. [17] whether T
˜
, P

˜
are pair−open covers,we say that P

˜
is a parallel re-

finement of T
˜
, solely in the event that any P1 ∈ P

˜
, in a way that P ∈ κ1is included

in T1 ∈ T
˜
and T1 ∈ κ1, and P2 ∈ V

˜
, such that P ∈ κ2 is included in T2 ∈ T

˜
and T2 ∈ κ2 .

Definition 7. [4] A pair−open cover P
˜

is known as locally finite particularly in the

event that ∀d ∈ D , There’s an open set T1 ∈ κ1, to the extent that T1 intersects numerous
individuals of P

⋂
κ1, or to the extent that an open set T2 ∈ κ2, to the extent that T1

intersects numerous individuals of P
⋂
κ2.

Definition 8. [13] A space (D,κ1 , κ2) is defined as pair−paracompact, whether and only
whether any pair−open cover has a pair−open locally finite parallel refinement.

Definition 9. [7] A point d of a space D is known as a condensation point of the set
M ⊂ D, if an arbitrary neigborhood (briefly, nbd) of the point d contains an uncountable
subset of this set.

Definition 10. [6] The intersection of countably several open sets is an open set when it
is the case unless space D is referred to by the term pair−space.

Definition 11. [9] Whenever each countably pair−open cover of a bitopological space
(D,κ1 , κ2) has a finite subcover, therefore the space is referred to be pair−countably com-
pact.

Definition 12. [9] When there is a finite subcover for each countably κ1κ2−open cover of
a bitopological space(D,κ1 , κ2), subsequently the space has been referred to as s−countably
compact.

Definition 13. [18] Whenever a function Υ : (D,κ1 , κ2) → (G, υ1 , υ2) is referred to
as pair−weakly continuous, it means that Υ−1(T ) is pair−ω−open for each pair−open set
T ⊂ G.

Definition 14. [14] Assuming a bitopological space (D,κ1 , κ2), we declare that κ1is lo-
cally Lindel

..
of with respect to κ2.When there is a κ1 nbd Td of d. In a way that Td

κ2 is
pair−Lindel

..
of all of them d ∈ (D,κ1 , κ2).



A. Atoom et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2574-2585 2577

3. A Novel Categorization Of Closed Functions

The notion of ω−closed functions in bitopological spaces is introduced and their rela-
tion to other spaces is illustrated in this section.

Definition 15. A pair−ω−closed function can be expressed as

Υ : (D,κ1 , κ2) → (G, υ1 , υ2)

when it mappings pair−closed sets onto pair−ω−closed sets.

Definition 16. A pair− semi−ω−closed function can be expressed as

Υ : (D,κ1 , κ2) → (G, υ1 , υ2)

when it mappings pair− semi closed sets onto pair−semi−ω−closed sets.

Definition 17. Whenever Υ−1(L) is pair− Lindel
..
of every individual pair− Lindel

..
of closed

subset L of (G, υ1 , υ2), subsequently

Υ : (D,κ1 , κ2) → (G, υ1 , υ2)

is a pair− Lindel
..
of function.

Definition 18. Whenever Υ−1(L) is pair−semi Lindel
..
of every individual pair− semi

Lindel
..
of closed subset L of (G, υ1 , υ2), subsequently

Υ : (D,κ1 , κ2) → (G, υ1 , υ2)

is a pair−semi Lindel
..
of function.

Definition 19. If the intersection of a countably many open sets is an ω−open set, there-
fore space D is known to as a P̋−space.

Definition 20. When there is a pair−open subset Td including d that means Td − J is
a countable set, therefore a subset J of a bitopological space (D,κ1 , κ2) is pair− ω−open.
Pair− ω−closed sets have been defined to be the complement of pair− ω−open sets.

Definition 21. Pair−ω − BO(J) as well as pair− ω − BC(J) is the family of all pair−
ω−open as well as pair− ω−closed subsets of a space (D,κ1 , κ2). Moreover, pair−ω −
BO(D; d) represents the family of all pair−ω−open sets of (D,κ1 , κ2) including d.

Definition 22. Whether there’s a κ1κ2−open subset Td comprising d that implies Td−J is
a countable set. Consequently a subset J of a bitopological space (D,κ1 , κ2) is pair−semi−
ω−open. Pair−semi-−ω−closed sets deemed to be the complement of pair−semi− ω−open
sets.

Definition 23. Pair−semi−ω−BO(J) as well as pair−semi− ω−BC(J) is the family of
all pair−semi− ω−open. Additionally pair−semi− ω−closed subsets of a space (D,κ1 , κ2).
Furthermore, pair−semi−ω − BO(D; d) represents the family of all pair−semi−ω−open
sets of (D,κ1 , κ2) encompassing d.



A. Atoom et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2574-2585 2578

Theorem 1. In a space (D,κ1 , κ2), any pair− Lindel
..
of, pair−ω−open subset J has the

form L\N , where L is a pair−open and N is a countable set; specifically, J is a Gδ−set.

Proof. For any value of d in J , there’s is a pair−open subset Td which includes d
and is countable set Td − J . A countable set is created when there is a pair−open subset
Td − J containing d for every d that is in J .

Claim T1, T2, ... ∈ κ1 , T
∗
1 , T

∗
2 , ... ∈ κ2 , thus J ⊂

∞⋃
i=1

Ti ∪
∞⋃
j=1

T ∗
j , during which Ti ∩ (J −D)

is κ1 countable,T ∗
j ∩ (J −D) is κ2−countable.

Presently, Ti ∩ (J −D) =
∞⋃
n=1

Di,n, i = 1, 2, ..., T ∗
j ∩ (J −D) =

∞⋃
m=1

Dh,m, h = 1, 2...

Right now, J = ∪(Ti\
∞⋃
n=1

Di,n) ∪ (T ∗
h\

∞⋃
m=1

Dh,m) = ∪
∞⋃
i=1

(Ti\L1) ∪
∞⋃
j=1

(T ∗
h\L2).

Enable L = L1 ∪ L2, and L ⊂
∞⋃
n=1

Di,n ∪
∞⋃

m=1
Dh,m.

Corollary 1. Let’s consider the hereditary Lindel
..
of space (D,κ1 , κ2). Following this, a

Gδ−set is any pair−ω−open subset of a space (D,κ1 , κ2).

Theorem 2. In a space (D,κ1 , κ2), any s− Lindel
..
of, s−ω−open subset J has the form

L\N , where L is a κ1κ2−open and N is a countable set; specifically, J is a Gδ−set.

Proof. The proof use the same methodology as theorem 1.

Corollary 2. Let’s consider the hereditary Lindel
..
of space (D,κ1 , κ2). Following this, a

Gδ−set is any s−ω−open subset of a space (D,κ1 , κ2).

It is incorrect to assert that theorem 3.1 is contradictory. Considering a specific illus-
tration:

Example 1. Consider two topologies κ1 , κ2 on R by the basis

H1 = {(−∞, j) : j > 0} ∪ {{d} : d > 0} , H2 = {(d,∞) : d < 0} ∪ {{d} : d < 0} ,

then (R, κ1 , κ2) is pair−Lindel
..
of .

The ensuing theorem extends the widely recognized theorem, which states that closed
continuous functionings with pair−Lindel

..
of counter images maintain the pair−Lindel

..
of

property under taking counter images.

Theorem 3. Letting Υ represent a pair−continuous pair−ω−closed functioning of a space
onto (G, υ1 , υ2) from (D,κ1 , κ2). which means that for every g ∈ (G, υ1 , υ2), Υ

−1(g) is
pair−Lindel

..
of . While (G, υ1 , υ2) is such a case, therefore (D,κ1 , κ2) is pair−Lindel

..
of.

Proof. Let us know T
˜

= {Tδ:δ ∈ Ξ} is a pair−open cover of (D,κ1 , κ2). In the

meantime late ∀g ∈ (G, υ1 , υ2), Υ−1(g) is pair−Lindel
..
of, the situation exists a countable



A. Atoom et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2574-2585 2579

subsets Ξy, Ξ
∗
y of Ξ , which means Υ−1(g) ⊆

⋃
δ∈Ξg

{Nδ : δ ∈ Ξg}
⋃ ⋃

α∈ Λ∗
y

{Mδ : δ ∈ Ξ∗
g},

at which {Nδ : δ ∈ Ξg} is κ1−open ,{Mδ : δ ∈ Ξ∗
g} is κ2−open.

Assume Hg = (G, υ1 , υ2)−Υ((D,κ1 , κ2)−
⋃

δ∈Ξg

Nδ) is a ρ1-open set comprising g,

and H∗
g = (G, υ1 , υ2)−Υ((D,κ1 , κ2)−

⋃
δ∈Ξ∗

g

Mδ) is a ρ2-open set comprising g, where

Υ−1( Hg) ⊆
⋃

δ∈Ξg

Vα, Υ−1 (H∗
g ) ⊆

⋃
δ∈Ξ∗

g

Mδ.

Assume H
˜ g

= {Hg : g ∈ (G, υ1 , υ2)}
⋃

{H∗
g : g ∈ (G, υ1 , υ2)} is a pair−open cover of

(G, υ1 , υ2). Considering Υ is pair−ω−closed, H
˜ g

is pair−ω−open for each g ∈ (G, υ1 , υ2).

Thus, there is an open pair−nbd H
\
g .In a manner that H

\
g ∩ ((D,κ1 , κ2)−H

\
g ) is

countable. Now H
\
g = (H

˜ g
∩Hg)∪ H

\
g ∩ ((D,κ1 , κ2)−Hg ).Consequently, Υ−1 (H

\
g ) is

enclosed in a union of countably large number of members of T
˜
.

Because of this
{
H

\
g , g ∈ (G, υ1 , υ2)

}
is a pair−open cover of (G, υ1 , υ2) and it is pair−Lindel

..
of,{

H
\
g , g ∈ (G, υ1 , υ2)

}
has a countable subcover. Therefore (D,κ1 , κ2) is the union of

countably large number of members of
{
Υ−1 (H

\
g ) , g ∈ (G, υ1 , υ2)

}
,since each Υ−1 (H

\
g ) is

contained in the union of countably large number of members of T
˜
.Consequently, (D,κ1 , κ2)

is the union of countably large number of members of T
˜
.Hence, (D,κ1 , κ2) is pair−Lindel

..
of .

Corollary 3. (i) A pair− ω−subset of a pair− Lindel
..
of space is pair−Lindel

..
of (ii) If

Υ : (D,κ1 , κ2) → (G, υ1 , υ2) is pair−continuous function from (D,κ1 , κ2) to (G, υ1 , υ2), so
the subsequent ones are comparable :

(a) Υ is pair−ω−closed ; (b) for each g ∈ (G, υ1 , υ2) and any pair−open set T , that
is to say Υ−1(g) ⊂ T, it actually exists a pair−ω−open set Qg such that g ∈ Qg and
Υ−1(Qg) ⊂ U.

Corollary 4. (i) A s− ω−subset of a s− Lindel
..
of space is s−Lindel

..
of (ii) If Υ :

(D,κ1 , κ2) → (G, υ1 , υ2) is s−continuous function from (D,κ1 , κ2) to (G, υ1 , υ2), so the
subsequent ones are comparable :(a) Υ is s−ω−closed ; (b) for each g ∈ (G, υ1 , υ2) and
any s−open set T , that is to say Υ−1(g) ⊂ T, it actually exists a s−ω−open set Qg such
that g ∈ Qg and Υ−1(Qg) ⊂ U.

Theorem 4. Assume Υ be pair−continuous s-ω−closed functioning of a space (D,κ1 , κ2)

onto (G, υ1 , υ2),so that Υ−1(g) is s-Lindel
..
of, for every g ∈ (G, υ1 , υ2), subsequently

(D,κ1 , κ2) is s-Lindel
..
of, whether (G, υ1 , υ2) is indeed.

Proof. Using the identical method as the theorem previously mentioned, the proof is
produced.

Theorem 5. Suppose that Υ is a ω−closed pair−continuous function of a



A. Atoom et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2574-2585 2580

regular space (D,κ1 , κ2) onto (G, υ1 , υ2).When (G, υ1 , υ2) is pair−paracompact and
Υ−1(g) is pair−paracompact relative to (D,κ1 , κ2). For every g in (G, υ1 , υ2), then (D,κ1 , κ2) is
pair−paracompact .

Proof. Present alongside T
˜

= {Tδ:δ ∈ Ξ} is a pair−open cover of (D,κ1 , κ2). Mean-

while, early ∀g ∈ (G, υ1 , υ2), Υ−1(g) is pair−paracompact, T
˜

has a pair−open locally

finite refinement in(D,κ1 , κ2) which at first cover Υ−1(g),It is a real issue a countable
subsets Ξg, Ξ

∗
g of Ξ , this implies Υ−1(g) ⊆

⋃
δ∈Ξg

{Nδ : δ ∈ Ξg}
⋃ ⋃

α∈ Λ∗
y

{Mδ : δ ∈ Ξ∗
g}, at

which {Nδ : δ ∈ Ξg} is κ1−open ,{Mδ : δ ∈ Ξ∗
g} is κ2−open.

Consider Hg = (G, υ1 , υ2)−Υ((D,κ1 , κ2)−
⋃

δ∈Ξg

Nδ) is a ρ1-open set comprising g,

and H∗
g = (G, υ1 , υ2)−Υ((D,κ1 , κ2)−

⋃
δ∈Ξ∗

g

Mδ) is a ρ2-open set comprising g, where

Υ−1( Hg) ⊆
⋃

δ∈Ξg

Vα, Υ−1 (H∗
g ) ⊆

⋃
δ∈Ξ∗

g

Mδ. Assume H
˜ g

= {Hg : g ∈ (G, υ1 , υ2)}⋃
{H∗

g : g ∈ (G, υ1 , υ2)} is a pair−open cover of (G, υ1 , υ2). Taking into account Υ
is pair−ω−closed, H

˜ g
is pair−ω−open for each g ∈ (G, υ1 , υ2). Thus, there is an open

pair−neibourhood H
\
g .In away that H

\
g ∩ ((D,κ1 , κ2) −H

\
g ) is countable. Consider-

ing (G, υ1 , υ2) is pair−paracompact H
˜

has pair−open locally finite parallel refinement de-

clare that:Q
˜
= {QD : D ∈ Ω1 }

⋃
{Q∗

D : D ∈ Ω2 }, where {QD : D ∈ Ω1 } is υ1-locally fi-

nite paracompact of Hg,and {Q∗
D : D ∈ Ω2} is υ2-locally finite paracompact of H

\
g , Ω =

Ω1
⋃

Ω2.Let L1 = {Υ−1(QD)
⋂

δi
Nδ , i = 1, 2, ..., n,D ∈ Ω1, δ ∈ Ξg} is κ1− open lo-

cally finite parallel refinement of {Nδ : δ ∈ Ξg},and let L2 = {Υ−1(Q∗
D)

⋂
Mδi , i =

1, 2, ..., n,D ∈ Ω2, δ ∈ Ξ∗
g} is κ2− open locally finite parallel refinement of {Mδ : δ ∈ Ξ∗

g}.
Let L

˜
= {L1

⋃
L2} , then L

˜
is pair−open locally finite parallel refinement of T

˜
, so

(D,κ1 , κ2) is pair−paracompact space.

Theorem 6. Allow Υ : (D,κ1 , κ2) → (G, υ1 , υ2) is pair−continuous function from
(D,κ1 , κ2)onto (G, υ1 , υ2), where (G, υ1 , υ2)is pair−locally Lindel

..
of pair−Hausdorff P̋−space.

Therefore the subsequent statements are comparable:

(a) Υ is a pair−ω−closed function and for each g ∈ (G, υ1 , υ2),Υ
−1(g) is pair−Lindel

..
of .

(b) Υ is a pair− Lindel
..
of function.

Proof. (a) → (b) originates using the identical method as in Theorem 3.
(b) → (a) : Let Υ : (D,κ1 , κ2) → (G, υ1 , υ2) be pair−continuous function pair−Lindel

..
of,

where (G, υ1 , υ2) is pair−locally Lindel
..
of pair−Hausdorff P̋−space. Demonstrating that

Υ is pair−ω−closed function is adequate. Let S1is closed in κ1 .Assume Υ(S1) is not
ω− closed in υ1 , therefor a point is present g0 ∈ (G, υ1 , υ2) − Υ(S1). In this way in or-
der for every neighborhood N of g0, N ∩ Υ(S1) is uncountable. Because of (G, υ1 , υ2) is
pair−locally Lindel

..
of, there exists υ1−neighborhoodM of g0, so thatM

υ2 is pair−Lindel
..
of .

Check now Υ(S1)∩M
υ2 is not pair− Lindel

..
of. While such is the case, it is evident that it is

pair−ω−closed, therefore there is a υ1− neighborhood K of g0. In a way that Υ(S1)∩K is



A. Atoom et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2574-2585 2581

countable, and these is not feasible. Presently M
υ2 is pair− Lindel

..
of, so Υ−1(M

υ2 ) is
pair−Lindel

..
of and S1 ∩ Υ−1(M

υ2 ) is pair−Lindel
..
of subset of (D,κ1 , κ2).Consequently

Υ(S1 ∩ Υ−1(M
υ2 )) = Υ(S1) ∩M

υ2 is pair−Lindel
..
of, it is paradoxical.Therefore Υ(S1) is

not ω− closed in υ1 . Comparative to S2 is closed in κ2 , Υ(S2) is not ω− closed in υ2 . Hence
Υ(S) is pair− ω− closed .

4. A novel applications of projection and product theorems

Here, we derive various applications of projection and product theorems for pair−Lindel
..
of,

pair−paracompact spaces using the findings from the preceding sections.

Theorem 7. Assume (D,κ1 , κ2) is a pair−Lindel
..
of space and (G, υ1 , υ2) be a P̋−space,

subsequent the projection p : (D × G, κ1 × υ1 , κ2 × υ2) → (G, υ1 , υ2) is pair− ω− closed
functions.

Proof. Let g ∈ (G, υ1 , υ2) and N̋= {kδ : δ ∈ Ξ} × {lδ : δ ∈ Ξ} be a (κ1 × υ1), (κ2 ×
υ2) open cover ofD ×G,where {kδ : δ ∈ Ξ} is pair−open cover of (D,κ1 , κ2) and {lδ : δ ∈ Ξ} is
pair−open cover of (G, υ1 , υ2). In a way that p−1(g) = Dg = D × {g} ⊂ N. For every
(d, g) ∈ D × {g} . Let Jd and Jg(D) be a pair−open neighborhood of (D,κ1 , κ2) and
(G, υ1 , υ2), such that (d, g) ∈ Jd × Jg(D) ⊂ U. Now {Jd : d ∈ D} is pair−open cover of

(D,κ1 , κ2). Consequently it has a countable subcover {Jdi}
∞
i=1 .Thus, D×{g} ⊂

∞⋃
i=1

Jdi ×

Jg(Di) ⊂ N̋. Let Wg =
∞⋂
i=1

Jg(Di) and W = {Wg : g ∈ G} , then D × {g} ⊂
∞⋃
i=1

Jdi ×

Gy ⊂N̋ and Wg is pair−ω− open set, as of late (G, υ1 , υ2) is a P̋−space. Consequently, for
every g ∈ (G, υ1 , υ2), there is pair−ω− open setWg such that g ∈ Wg and p−1(g) ⊂N̋. Thus,
according to Theorem 1, the projection p is a pair− ω− closed functions.

Theorem 8. Assume (D,κ1 , κ2) is a s−Lindel
..
of space and (G, υ1 , υ2) be a P̋−space,

subsequent the projection p : (D × G, κ1 × υ1 , κ2 × υ2) → (G, υ1 , υ2) is s− ω− closed
functions.

Proof. The proof use the same methodology as Theorem 7.

Theorem 9. Let (D,κ1 , κ2), (G, υ1 , υ2) be any bitopological spaces,(D,κ1 , κ2) be a pair−Lindel
..
of

space and (G, υ1 , υ2) be a P̋−space then the projection function

π : (D ×G, κ1 × υ1 , κ2 × υ2) → (G, υ1 , υ2)

is pair− ω−closed.

Proof. If (D,κ1 , κ2) is pair−Lindel
..
of, then (D,κ1) is Lindel

..
of and (D,κ2) is Lindel

..
of.

Consequently, the projection functions: π1 : (D ×G, κ1 ×υ1) → (G, υ1), π2 : (D ×G, κ2 ×
υ2) → (G, υ2) are ω− closed. Thus π is pair−ω−closed.



A. Atoom et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2574-2585 2582

Theorem 10. Let (D,κ1 , κ2), (G, υ1 , υ2) be any bitopological spaces, (D,κ1 , κ2) be a
s−Lindel

..
of space and (G, υ1 , υ2) be a P̋−space then the projection function

π : (D ×G, κ1 × υ1 , κ2 × υ2) → (G, υ1 , υ2)

is s− ω−closed.

Proof. If (G, υ1 , υ2) is s−Lindel
..
of, then (D,κ1) is Lindel

..
of and (D,κ2) is Lindel

..
of.

Consequently, the projection functions: π1 : (D ×G, κ1 ×υ1) → (G, υ1), π2 : (D ×G, κ2 ×
υ2) → (G, υ2) are ω− closed. Thus π is s−ω−closed.

Theorem 11. Assume (G, υ1 , υ2) be a topological space in which a Fσ−set which is not
pair−ω−closed, and (D,κ1 , κ2) be any bitopological space. If the projection

(D ×G, κ1 × υ1 , κ2 × υ2) → (G, υ1 , υ2)

is pair− ω−closed, then (D,κ1 , κ2) is pair−countably compact.

Proof. Assume
∞⋃
i=1

Ji is a pair−F−subset of (G, υ1 , υ2) which is not pair− ω−closed,

and (D,κ1 , κ2) is not pair−countably compact. Subsequently, there a decreasing pairwise

sequence {Ki}∞i=1of pair−closed subsets of (D,κ1 , κ2), in a manner that
∞⋂
i=1

Ki = ϕ.

Let F =
∞⋃
i=1

(Ji×Ki, κ1 ×υ1 , κ2 ×υ2), afterwards it is evident to us that F is a pair−closed

subset of (J ×K,κ1 × υ1 , κ2 × υ2). Likewise for each of the points (d, g) ∈ (J ×K,κ1 ×
υ1 , κ2 × υ2), p(d, g) = g. Next p(F ) =

∞⋃
i=1

Ji is not pair−ω−closed, consequently the

projection is not pair− ω−closed.Thus, the outcome.

Corollary 5. Assume (G, υ1 , υ2) be a topological space in which a Fσ−set which is not
s−ω−closed, and (D,κ1 , κ2) be any bitopological space. If the projection

(D ×G, κ1 × υ1 , κ2 × υ2) → (G, υ1 , υ2)

is s− ω−closed, then (D,κ1 , κ2) is s−countably compact.

Theorem 12. A space (G, υ1 , υ2) is P̋−space if and only if for pair−Lindel
..
of space

(D,κ1 , κ2), then the projection (D ×G, κ1 ×υ1 , κ2 ×υ2) → (G, υ1 , υ2) is pair−ω−closed.

Proof. The requirement portion is derived from theorem 7, as the condition must be suf-
ficient. Presume (G, υ1 , υ2) is not P̋−space. For any pair−Lindel

..
of space (D,κ1 , κ2) then

the projection (D × G, κ1 × υ1 , κ2 × υ2) → (G, υ1 , υ2) is pair− ω−closed. Let D = R
is the set of real numbers with usual topology (R, κu, κu). Hence by the earlier theorem,
(D,κ1 , κ2) is pair−coutably compact, it is paradoxical.



A. Atoom et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2574-2585 2583

Theorem 13. Assume (D,κ1 , κ2), (G, υ1 , υ2) is any bitopological spaces with the property
that every pair−Lindel

..
of subset is pair−ω−closed. when Υ : (D,κ1 , κ2) → (G, υ1 , υ2) is

pair−Lindel
..
of, then Υ is pair−weakly continuous .

Proof. Let p1 : (D ×G, κ1 ×υ1 , κ2 ×υ2) → (D,κ1 , κ2), p2 : (D ×G, κ1 ×υ1 , κ2 ×υ2) →
(G, υ1 , υ2) be the projections, then (D,κ1 , κ2) and range Υ of a pair−Lindel

..
of set as images

of pair−Lindel
..
of sets under p1and p2.Let Z∗

1 = p1\Υ. Ensure that Z∗
1 is pair−ω−closed.

Actually, if T is pair−closed, then T is pair−Lindel
..
of, Z∗

1 (T ) is pair−Lindel
..
of. Therefore,

it is pair−ω−closed. Since Υ is a function defined on (D,κ1 , κ2), Z∗
1 is a bijection onto

(D,κ1 , κ2).This combined with reality that Z∗
1 is pair−ω−closed, means that for each

pair−open set V,Z∗
1 (V ) is pair−ω−open in (D,κ1 , κ2). Presently Υ = p2 ◦ Z∗−1

1 .Υ thus
possesses the necessary attribute.

Corollary 6. Let (D,κ1 , κ2) be a pair−Lindel
..
of space and (G, υ1 , υ2) be a P̋−space.

Therefore the subsequent statement is true:

(i) (D ×G, κ1 × υ1 , κ2 × υ2) is pair−Lindel
..
of if and only if (G, υ1 , υ2) is indeed,

(ii) (D ×G, κ1 × υ1 , κ2 × υ2) is pair−paracompact if and only if (G, υ1 , υ2) is so.

5. Some Counter Examples

We go over a number of counterexamples in this section that are pertinent to the
definitions and theorems in the preceding sections. We will begin with some instances
pertaining to the pair−ω− closed functions.

Example 2. Let Υbe functioning from a discrete countable space (D,κ1 , κ2) onto the
space of rationals (G, υ1 , υ2).Next, Υis a pair−continuous pair−ω−closed function. But
still Υ is not pair−closed. Additionally, for every g in (G, υ1 , υ2),Υ

−1(g) is pair−Lindel
..
of.

Additionally (D,κ1 , κ2), (G, υ1 , υ2) are a pair−Lindel
..
of spaces, therefore pair−paracompact

spaces. Theorem 3 is therefore more generic than the one that presumes the function to
be pair−closed.

In connection theorem 13, the example that follows is examined.

Example 3. Assume S be the sorgenfry line and the sorgenfry plane S × S. It is aware
of this (R, κs, κs) is pair−Lindel

..
of spaces, therefore pair−paracompact spaces. However

S × S is not pair−normal so it is not pair−paracompact .

Example 4. We are going to concentrate on P̋−space. Pay attention to any space lacking
a condensation point is a P̋−space, but not a pair−space. Considering the foregoing, any
countable space is a P̋−space. As an illustration of uncountable P̋−space N ∪ R , that is
first countable, locally compact and 0−dimensional, however, it lacks condensation point.
Consequently it is P̋−space though not a pair−space.



REFERENCES 2584

6. Conclusions

This study has shown us that the pair −ω−closed functions are an extension of
pair−closed functions. They are specified on topological spaces and have an effective
method of holding onto sequence bounds. This suggests that if a series has a sequence
of points in the function’s domain that converge to a point, then the image of the series
under the function will also converge to the image of the point. It’s a way to extend the
notion of closest to more complicated situations, such weakening these functions, therefore
we obtain and investigate their key characteristics in this study, to ensure the concepts of
pairwise pair−ω−closed are understood. We have examined the salient features of these
concepts and shown how they apply to different situations. We determined their overall
fundamental features and the prerequisites that need to be satisfied for similar linkages to
be made between them. We discussed their key characteristics and gave examples of how
they complement one another. The study provided multiple examples of various functions
along with highlighting their properties.These functions will act as a basis for additional
studies into the potential applications of each of these functions. Other versions of these
duties including fuzzy, soft, and group, [1],[10],[12],[16],[18] and [19]. , might be the subject
of future investigation.

Acknowledgements

We sincerely thank everyone who made a contribution to this research. Their cooper-
ation, guidance, and support have been crucial to completing the work.

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