Ratio Mathematica Volume 38, 2020, pp. 223-236

Pairwise Paracompactness

Pallavi S. Mirajakar*
P. G. Patil†

Abstract

The purpose of this paper is to introduce and study a new
paracompactness in bitopological spaces using (τi,τj)-g∗ωα-closed
sets. Further, the properties of (τi,τj)-g∗ωα-closed sets, (τi,τj)-g∗ωα-
continuous functions and (τi,τj)-g∗ωα-irresolute maps and (τi,τj)-
g∗ωα-paracompact spaces are discussed in bitopological spaces.
Keywords: (τi,τj)-g∗ωα-closed sets, (τi,τj)-g∗ωα-open sets,
(τi,τj)-g∗ωα-continuous and (τi,τj)-g∗ωα-irresolute maps,
(τi,τj)-g∗ωα-paracompact spaces.
2010 AMS subject classifications: 54E55. 1

*Department of Mathematics (P. C. Jabin Science College, Hubbali,Karnatak, India); psmira-
jakar@gmail.com.

†Department of Mathematics (Karnatak University, Dharwad, Karnatak, India); pg-
patil@kud.ac.in

1Received on January 10th, 2020. Accepted on May 3rd, 2020. Published on June 30th, 2020.
doi: 10.23755/rm.v38i0.495. ISSN: 1592-7415. eISSN: 2282-8214. ©Pallavi S. Mirajakar et al.
This paper is published under the CC-BY licence agreement.

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P. S. Mirajakar and P. G. Patil

1 Introduction
The research in topology over last two decades has reached a high level

in many directions. Topological methods are widely used in many other branches
of modern mathematics such as differential equation, functional analysis, classical
mechanics, general theory of relativity, mathematical economics, quantum theory,
biology etc.

Bitopological space is a triplet (X,τ1,τ2), where X is a non empty set and
τ1 and τ2 are topologies on a space X. In 1963, J. C. Kelly [8] initiated the study
of bitopological spaces. In 1985, Fututake [5] studied the concept of generalized
closed (briefly g-closed) sets in bitopological spaces. After that, several authors
turned their attention towards the generalizations of various concepts in topology
by considering bitopological spaces.

In this paper, (τi,τj)-g∗ωα-closed sets, (τi,τj)-g∗ωα-continuous functions
and (τi,τj)-g∗ωα-irresolute maps are defined and studied in bitopological spaces.
Also, the concept of (τi,τj)-g∗ωα-paracompactness in bitopological spaces is in-
troduced and studied.

2 Preliminaries
Throughout this present paper, let X, Y and Z always represents non-empty

bitopological spaces (X,τ1,τ2), (Y,σ1,σ2) and (Z,γ1,γ2) on which no separation
axioms are assumed unless explicitly mentioned and the integers i,j,k ∈{1, 2}.

Definition 2.1. [13] A space X is said to be g∗ωα-paracompact if every open
cover of X has a g∗ωα-locally finite g∗ωα-refinement.

Definition 2.2. Let A ⊆ X. Then A is said to be a
(a) ωα-closed [2] if αcl(A) ⊆ U whenever A ⊆ U and U is ω-open in X.
(b) g∗ωα-closed[12] if cl(A) ⊆ U whenever A ⊆ U and U is ωα-open in X.

Definition 2.3. A subset A of a bitopological space (X,τ1,τ2) is called a
(a) (τi,τj)-g-closed [5] if τj-cl(A) ⊆ U whenever A ⊆ U and U is open in τi.
(b) (τi,τj)-rg-closed [1] if τj-cl(A) ⊆ U whenever A ⊆ U and U is regular open
in τi.
(c) (τi,τj)-αg-closed [3] if τj-αcl(A) ⊆ U whenever A ⊆ U and U is open in τi.
(d) (τi,τj)-gα-closed [3] if τj-αcl(A) ⊆ U whenever A ⊆ U and U is α-open in
τi.
(e) (τi,τj)-gpr-closed [7] if τj-pcl(A) ⊆ U whenever A ⊆ U and U is regular
open in τi.
(f) (τi,τj)-g∗-closed [14] if τj-cl(A) ⊆ U whenever A ⊆ U and U is g-open in τi.

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(g) (τi,τj)-ωα-closed [11] if τj-cl(A) ⊆ U whenever A ⊆ U and U is ω-open in
τi.
In all the above definitions i 6= j.

Definition 2.4. A map f : (X,τ1,τ2) → (Y,µ1,µ2) is called a
(a) τj-µk-continuous [10] if f−1(G) ∈ τj for every open set G in µk.
(b) D(τi,τj)-µk-continuous [10] if the inverse image of every µk-closed set in
(Y,µ1,µ2) is (τi,τj)-g-closed in (X,τ1,τ2).
(c) Dr(τi,τj)-µk-continuous [1] if the inverse image of every µk-closed set in
(Y,µ1,µ2) is (τi,τj)-rg-closed in (X,τ1,τ2).
(d) C(τi,τj)-µk-continuous [6] if the inverse image of every µk-closed set in
(Y,µ1,µ2) is (τi,τj)-ω-closed in (X,τ1,τ2).
(e) D∗(τi,τj)-µk-continuous [14] if the inverse image of every µk-closed set in
(Y,µ1,µ2) is (τi,τj)-g∗-closed in (X,τ1,τ2).
(f) (τi,τj)-αg-continuous [4] if the inverse image of every µk-closed set in (Y,µ1,µ2)
is (τi,τj)-αg-closed in (X,τ1,τ2).

Definition 2.5. [8] A bitopological space (X,τ1,τ2) is said to be pairwise Haus-
dorff if for each pair of distinct points x and y of X, there exist U ∈ Pi and V ∈ Pj
such that x ∈ U, y ∈ V and U ∩V = φ.

3 (τi,τj)-g∗ωα-Closed Sets
This section deals with the concept of g∗ωα-closed sets in bitopological

spaces and some of their properties.

Definition 3.1. Let (i,j) ∈ {1, 2} where i 6= j. A subset A of a bitopological
space (X,τ1,τ2) is said to be (τi,τj)-g∗ωα-closed if τj-cl(A) ⊆ U whenever
A ⊆ U and U ∈ τi-ωα-open in X.

Example 3.1. Let X = {m,n,p}, τ1 = {X,φ,{m},{n,p}} and τ2 = {X,φ,{m}}.
Consider a set in the space (X,τ1,τ2), A = {n,p} which is (τ1,τ2)-g∗ωα-closed.

Remark 3.1. If τ1 = τ2 = τ in Definition 3.1, then (τi,τj)-g∗ωα-closed set in
(X,τ1,τ2) is same as g∗ωα-closed [12] in (X,τ).

The family of all (τi,τj)-g∗ωα-closed sets in (X,τ1,τ2) is denoted by P(τi,τj).

Theorem 3.1. Every τj-closed (resp. (τi,τj)-regular closed) is (τi,τj)-g∗ωα-
closed.
However the converse need not be true in general as shown in the following ex-
ample.

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Example 3.2. Let X = {m,n,p}, τ1 = {X,φ,{m},{n,p}} and τ2 = {X,φ,{m}}.
Consider the set, A = {m,p} is (τ1,τ2)-g∗ωα-closed but not τ2-closed (resp.
(τi,τj)-regular closed).

We have the following implification:
τj-closed → (τi,τj)-g∗ωα-closed → (τi,τj)-αg-closed

Remark 3.2. If A and B are (τi,τj)-g∗ωα-closed in (X,τ1,τ2) then A∪B is also
(τi,τj)-g∗ωα-closed.

Theorem 3.2. If a subset A of (X,τ1,τ2) is (τi,τj)-g∗ωα-closed then τj-cl(A)−A
does not contain any non empty ωα-closed set in τi.

Proof. Let A ⊆ (τi,τj)-g∗ωα-closed and F ⊆ τi-ωα-closed set such that
F ⊆ τj-cl(A) −A. Now F ⊆ τj-cl(A) and F ⊆ X −A. Then A ⊆ X −F and
by hypothesis A is (τi,τj)-g∗ωα-closed and X − F is τi − ωα-open. Thus from
Definition 3.1, τj-cl(A) ⊆ X − F , that is F ⊆ (X − τj-cl(A)). Then F ⊆ (τj-
cl(A)) ∩ (X − τj-cl(A)) = φ and so F = φ which is a contradiction. Hence
τj-cl(A) −A does not contain any non empty ωα-closed set. 2

Remark 3.3. A (τi,τj)-g∗ωα-closed set need not be τi-g∗ωα-closed or τj-g∗ωα-
closed.

Example 3.3. Let X = {m,n,p}, τ1 = {X,φ,{m}} and τ2 = {X,φ}. Then the
set A = {n,p} is (τi,τj)-g∗ωα-closed but not τ2-g∗ωα-closed.
Also, if X = {m,n,p}, τ1 = {X,φ,{p}} and τ2 = {X,φ,{m},{m,n}} be
topology on X. Then the set A = {m,p} is(τi,τj)-g∗ωα-closed but not τ1-g∗ωα-
closed in (X,τ1,τ2).

Remark 3.4. In general P(τi,τj) 6= P(τj,τi).

Example 3.4. Let X = {m,n,p}, τ1 = {X,φ,{p}} and τ2 = {X,φ,{m},{m,n}}.
Then g∗ωαC(τ1,τ2) = {X,φ,{m,n}} and g∗ωαC(τ2,τ1) = {X,φ,{n,p},{p}}.
Hence we can observe that g∗ωαC(τ1,τ2) 6= g∗ωαC(τ2,τ1).

Remark 3.5. If τ1 ⊆ τ2, then P(τ2,τ1) ⊆ P(τ1,τ2) but converse is not true.

Example 3.5. Let X = {m,n,p}, τ1 = {X,φ,{p}} and τ2 = {X,φ,{m},{m,n}}.
Then P(τ2,τ1) = {X},φ,{m,n},{n,p}} and P(τ1,τ2) = {X,φ,{m,n},{n,p},{p}}.
Then P(τ2,τ1) ⊆ G(τ1,τ2) but τ1 * τ2.

Theorem 3.3. A τi-ωα-open and (τi,τj)-g∗ωα-closed set is τj-closed.

Proof. Now A ⊆ A. Then τj-cl(A) ⊆ A and A ⊆ τj-cl(A). Therefore
τj-cl(A) = A and hence A ∈ τj-closed. 2

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Theorem 3.4. Let A be τi-ωα-open and (τi,τj)-g∗ωα-closed. Suppose F is τj-
closed, then A∩F is (τi,τj)-g∗ωα-closed.

Proof. Let A be τi-ωα-open and A be (τi,τj)-g∗ωα-closed and F be τj-closed.
Then from Theorem 3.3, A is τj-closed. So A∩F is τj-closed and hence (τi,τj)-
g∗ωα-closed. 2

Theorem 3.5. If A is (τi,τj)-g∗ωα-closed and A ⊆ B ⊆ τj-cl(A), then τj-cl(B)−
B contains no non empty τi-closed set.

Proof. Let A be (τi,τj)-g∗ωα-closed and A ⊆ B ⊆ τj-cl(A). Then B is
(τi,τj)-g∗ωα-closed follows from Theorem 3.19 [12]. Hence τj-cl(B) − B con-
tains no non empty τi-closed set. 2

Corolary 3.1. If A is (τi,τj)-g∗ωα-closed and A ⊆ B ⊆ τj-cl(A), then τj-cl(B)−
B contains no non empty τi-ωα-closed set.

Theorem 3.6. Arbitrary union of (τi,τj)-g∗ωα-closed sets {Ai : i ∈ I} is (τi,τj)-
g∗ωα-closed if the family {Ai : i ∈ I} is τj-locally finite.

Proof. Let {Ai : i ∈ I} is τj-locally finite and {Ai : i ∈ I} is (τi,τj)-
g∗ωα-closed. Let ∪Ai ⊆ U where U ∈ τi-ωα-open. Then Ai ⊆ U and ωα-
open in τi. Since A is (τi,τj)-g∗ωα-closed the for each i ∈ I, τj-cl(Ai) ⊆ U.
Consequently ∪τj-cl(Ai) ⊆ U. Since the family, {Ai : i ∈ I} is τj-locally finite
τj-cl(∪Ai) = ∪(τj-cl(Ai)) ⊆ U. Therefore ∪Ai is (τi,τj)-g∗ωα-closed. 2

Theorem 3.7. For an element x in X, the set X −{x} is (τi,τj)-g∗ωα-closed or
X −{x} is ωα-open in τi.

Proof. Suppose X −{x} is not ωα-open in τi, then X is the only ωα-open
set containing X −{x}, that is τj-cl(X −{x}) ⊆ τj-cl({x}) = X. Hence τj-
cl(X −{x}) ⊆ X. Thus X −{x} is τi,τj)-g∗ωα-closed. 2

Definition 3.2. A subset A of a bitopological space (X,τ1,τ2) is (τi,τj)-g∗ωα-
open if its complement is (τi,τj)-g∗ωα-closed.

Definition 3.3. For a subset A of a bitopological space (X,τ1,τ2), (τi,τj)-g∗ωα-
interior of A is denoted by (τi,τj)-g∗ωα-int(A) and is defined as
(τi,τj)-g∗ωα-int(A) = ∪{F : F ∈ (τi,τj)-g∗ωα-open and F ⊆ A}.

Theorem 3.8. Let A be (τi,τj)-g∗ωα-open. Then P = X whenever G is τi-ωα-
open and τj-g∗ωα-int(A) ∪Ac ⊆ G.

Theorem 3.9. A set A is (τi,τj)-g∗ωα-open if and only if F ⊆ τj-int(A) whenever
F is τi-closed and F ⊆ A.

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Theorem 3.10. If A and B are separated (τi,τj)-g∗ωα-open sets then A ∪ B is
also (τi,τj)-g∗ωα-open.

Proof.Suppose A and B are (τi,τj)-g∗ωα-open sets. Let F be an τi-closed set
such that F ⊆ A∪B. Since A and B are separated, τi-cl(A) ∩B = A∩ τi-cl(B)
= φ and τj-cl(A) ∩B = A∩ τj-cl(B) = φ. Then F ∩ τj-cl(A) ⊆ (A∪B) ∩ τj-
cl(A) = A. Similarly, F ∩ τj-cl(B) ⊆ B. Since F is τi-closed, we have F ∩ τi-
cl(A),F ∩ τi-cl(B) are also τi-closed and from hypothesis A and B are (τi,τj)-
g∗ωα-open sets, F ∩ τj-cl(A) ⊆ τj-int(A) and F ∩ τj-cl(B) ⊆ τj-int(B). Now
F = F ∩ (A∪B) ⊆ (F ∩ τj-cl(A)) ∪ (F ∩ τj-cl(B)) ⊆ τj-int(A∪B). Hence
A∪B is (τi,τj)-g∗ωα-open. 2

Definition 3.4. A bitopological space (X,τ1,τ2) is said to be a (τi,τj)-Tg∗ωα-
space if every (τi,τj)-g∗ωα-closed set is τj-closed.

Example 3.6. Let X = {m,n,p}, τ1 = {X,φ,{m},{m,n}} and τ2 = {X,φ,{m},
{p},{m,p}}. Then (X,τ1,τ2) is (τ1,τ2)-Tg∗ωα-space.

Theorem 3.11. If a bitopological space (X,τ1,τ2) is (τi,τj)-Tg∗ωα space, then
for each x ∈ X, {x} is τi-ωα-closed or τj-open.

Proof. Suppose {x} is not (τi,τj)-g∗ωα-open, then {x}c is (τi,τj)-g∗ωα-
closed. As X is is (τi,τj)-Tg∗ωα-space, {x}c is τj-closed and hence {x} is τj-open.
2

Remark 3.6. Every singleton subset of (X,τ1,τ2) is τj-closed or τi-ωα-closed
but (X,τ1,τ2) is not (τi,τj)-Tg∗ωα-space.

Example 3.7. Let X = {m,n,p}, τ1 = {X,φ,{m},{m,n}} and τ2 = {X,φ,{m},
{p},{m,p}}. Then every singleton set {x} of X is either τ2-open or τ1-ωα-closed.
However, (X,τ1,τ2) is not (τ1,τ2)-Tg∗ωα-space.

Remark 3.7. If (X,τ1) and (X,τ2) are both Tg∗ωα-space, then it need not imply
(τ1,τ2)-Tg∗ωα-space.

Example 3.8. Let X = {m,n,p}, τ1 = {X,φ,{n},{n,p}} and τ2 = {X,φ,{m},
{m,n}}. Then (X,τ1) and (X,τ2) are Tg∗ωα-space, but (X,τ1,τ2) is not (τ1,τ2)-
Tg∗ωα-space.

Remark 3.8. The space (X,τ1) is not generally Tg∗ωα-space if (X,τ1,τ2) is (τ1,τ2)-
Tg∗ωα-space.

Example 3.9. Let X = {m,n,p}, τ1 = {X,φ,{m},{n,p}} and τ2 = {X,φ,{m}}.
Then (X,τ1,τ2) is (τ1,τ2)-Tg∗ωα-space, but (X,τ1) is not Tg∗ωα-space.

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4 (τi,τj)-g∗ωα-Continuous and (τi,τj)-g∗ωα-Irresolute
Maps

Definition 4.1. A map f : (X,τ1,τ2) → (Y,µ1,µ2) is called P(τi,τj)-µk-continuous
(pairwise g∗ωα-continuous) if the inverse image of every µk-closed set in (Y,µ1,µ2)
is (τi,τj)-g∗ωα-closed in (X,τ1,τ2).

Theorem 4.1. Every is τj-µk-continuous function is P(τi,τj)-µk-continuous.

Proof. Follows from Theorem 3.1. 2
The converse need not be true as seen from the following example.

Example 4.1. Let X = Y = {m,n,p}, τ1 = {X,φ,{m},{n,p}}, τ2 = {X,φ,
{m},{p},{m,p}}, µ1 = {Y,φ,{n}} and µ2 = {Y,φ,{m}}. Let f : (X,τ1,τ2) →
(Y,µ1,µ2) be the identity map. Then f is P(τ1,τ2)-µ1-continuous but not τ2-µ1-
continuous, since for the µ1-closed set A = {m,p} in Y, f−1({m,p}) = {m,p}
is not τ2-closed in X.

Remark 4.1. Let f : (X,τ1,τ2) → (Y,µ1,µ2) be P(τi,τj)-µk-continuous g :
(Y,µ1,µ2) → (Z,γ1,γ2) be P(µ1,µ2)-γm-continuous but their composition need
not be P(τi,τj)-γm-continuous.

Example 4.2. Let X = Y = {m,n,p}, τ1 = {X,φ,{m},{m,n}}, τ2 = {X,φ,
{m},{p},{m,p}}, µ1 = {Y,φ,{m},{n,p}}, µ2 = {Y,φ,{m}}, γ1 = {Z,φ,
{m},{m,p}} and γ2 = {Z,φ,{m},{m,n},{m,p}}. Let f : (X,τ1,τ2) →
(Y,µ1,µ2) be identity map and define a map g : (Y,µ1,µ2) → (Z,γ1,γ2) by
g(m) = n,g(n) = m,g(p) = p. Then f and g are pairwise g∗ωα-continuous maps
but their composition is not pairwise g∗ωα-continuous, since for the γ1-closed set
{n,p} in (Z,γ1,γ2), (gof)−1({n,p}) = f−1(g−1({n,p})) = f−1({m,p}) =
{m,p} is not (τ1,τ2)-g∗ωα-closed in (X,τ1,τ2).

Definition 4.2. A map f : (X,τ1,τ2) → (Y,µ1,µ2) is called pairwise g∗ωα-
irresolute if for every A ∈ P(µk,µe) in (Y,µ1,µ2), f−1(A) ∈ P(τi,τj) in (X,τ1,τ2).

Theorem 4.2. If a map f : (X,τ1,τ2) → (Y,µ1,µ2) pairwise g∗ωα-irresolute if f
is P(τi,τj)-µe-continuous.

Proof. Let F be µe-closed, then F is (µk,µe)-g∗ωα-closed in (Y,µ1,µ2). From
Theorem 3.1, F ∈ P(µk,µe). Since f is pairwise g∗ωα-irresolute, f−1(F) ∈
P(τi,τj). Therefore f is P(τi,τj)-µe-continuous. 2
The converse of this theorem need not be true as seen from the following example.

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P. S. Mirajakar and P. G. Patil

Example 4.3. Let X = Y = {m,n,p}, τ1 = {X,φ,{m},{m,n}}, τ2 = {X,φ,
{m},{p},{m,p}}, µ1 = {Y,φ,{n},{n,p}} and µ2 = {Y,φ,{m},{m,p}}.
Let f : (X,τ1,τ2) → (Y,µ1,µ2) be the identity map. Then f is P(τ1,τ2)-µ1-
continuous map but not pairwise g∗ωα-irresolute map, since for the (µ1,µ2)-
g∗ωα-closed set {m,p} in (Y,µ1,µ2), f−1({m,p}) = {m,p} is not (τi,τj)-g∗ωα-
closed set in (X,τ1,τ2).

Theorem 4.3. Let f : (X,τ1,τ2) → (Y,µ1,µ2) be a map and (Y,µ1,µ2) be
(µk,µe)-Tg∗ωα-space. Then f is pairwise g∗ωα-irresolute if and only if f is P(τi,τj)-
µe-continuous.

Proof. Suppose f is pairwise g∗ωα-irresolute. From Theorem 4.2, f is P(τi,τj)-
µe-continuous.
Conversely, let f be P(τi,τj)-µe-continuous map. Let F be (µk,µe)-g∗ωα-closed
in (Y,µ1,µ2). By hypothesis (Y,µ1,µ2) is (µk,µe)-Tg∗ωα-space, F is µe-closed set
in (Y,µ1,µ2). Again, since f is P(τi,τj)-µe-continuous, f−1(F) is (τi,τj)-g∗ωα-
closed set in (X,τ1,τ2). Hence f is pairwise g∗ωα-irresolute. 2

5 (τi,τj)-g∗ωα-Paracompact Spaces
We recall that, a collection ξ = {Fλ : λ ∈ Γ} of subsets of a space X is called

a locally finite with respect to the topology τi, if for each x ∈ X there exists
Ux ∈ τi containing x and Ux which intersects at most finitely many members of ξ.

Definition 5.1. A collection ξ = {Fλ : λ ∈ Γ} of subsets of a space X is called
(τi,τj)-P-locally finite if for each x ∈ X there exist (τi,τj)-g∗ωα-open Ux in X
and Ux intersects at most finitely many members of ξ.

Theorem 5.1. Let ξ = {Fλ : λ ∈ Γ} be a collection of subsets of (X,τ1,τ2) then
(a) ξ is (τi,τj)-g∗ωα-locally finite if and only if {(τi,τj)g∗ωα-cl(Fλ) : λ ∈ Γ} is
(τi,τj)-P-locally finite.
(b) if ξ is (τi,τj)-P-locally finite, then ∪(τi,τj)g∗ωα-cl(Fλ) = (τi,τj)g∗ωα-cl(∪Fλ).
(c) ξ is locally finite with respect to the topology τi if and only if the collection
{(τi,τj)g∗ωα-cl(Fλ : λ ∈ Γ)} is locally finite with respect to the topology τi.

Proof. (a) Suppose ξ is (τi,τj)-P-locally finite. Then for each x ∈ X, there
exists (τi,τj)-g∗ωα-open set Ux containing x, which meets only finitely many of
the sets Fλ, say Fλ1,Fλ2, ...,Fλn . Since FλK ⊆ (τi,τj)g

∗ωα-cl(FλK ) for each k
= 1,2,...,n and Ux meets (τi,τj)-g∗ωα-cl(Fλ1 ), ..., (τi,τj)g

∗ωα-cl(Fλn ). Therefore
g∗ωα-cl(Fλ) where λ ∈ γ is (τi,τj)-P-locally finite.
Conversely, let x ∈ X. Then there exists (τi,τj)-g∗ωα-open Ux, which meets only
finitely many of the sets (τi,τj)-g∗ωα-cl(Fλn ), say (τi,τj)-g

∗ωα-cl(Fλ1 ), ...,

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(τi,τj)g
∗ωα-cl(Fλn ). Then Ux ∩ (τi,τj)-g∗ωα-cl(Fλk ) 6= φ. Let q ∈ Ux and

q ∈ (τi,τj)-g∗ωα-cl(Fλk ), implies that for every (τi,τj)-g
∗ωα-open set Vq, we

have Vq ∩Fλk 6= φ. But, we have Ux is (τi,τj)-g
∗ωα-open set containing q and so

Ux ∩Fλk 6= φ for each k=1,2,...,n. Thus ξ is (τi,τj)-P locally finite.
(b) Suppose ξ is (τi,τj)-P-locally finite, then ∪(τi,τj)-g∗ωα-cl(Fλ) ⊆ (τi,τj)g∗ωα-
cl(∪Fλ). On the other hand, let q ∈ (τi,τj)-g∗ωα-cl(∪Fλ). Then for every (τi,τj)-
g∗ωα-open set Vq such that Vq ∩(∪Fλ) 6= φ. But from the hypothesis, there exists
(τi,τj)-g∗ωα-open set Uq such that Uq meets only finitely many of the sets Fλ, say
Fλ1,Fλ2, ...,Fλn . Thus for each (τi,τj)-g

∗ωα-open set Vq containing q, we have
Vq ∩ (∪Fλk ) 6= φ where k=1,2,...,n. That is, for each q ∈ (τi,τj)-g

∗ωα-cl(∪Fλk ),
there exists h such that q ∈ (τi,τj)-g∗ωα-cl(Fλh ). Therefore q ∈ ∪(τi,τj)-g

∗ωα-
cl(Fλ) and hence (τi,τj)-g∗ωα-cl(∪Fλ) = ∪(τi,τj)-g∗ωα-cl(Fλ).
(c) Suppose ξ is locally finite with respect to the topology τi, then for each x ∈ X
there exists τi-open set Ux which meets only finitely many set Fλ, say Fλ1,Fλ2, ...
,Fλn , but Fλk ⊆ (τi,τj)-g∗ωα-cl(Fλk ). Then Ux meets (τi,τj)-g

∗ωα-cl(Fλ1 ), ...,
(τi,τj)-g∗ωα-cl(Fλn ). Thus (τi,τj)-g

∗ωα-cl(Fλ : λ ∈ Γ) is locally finite with
respect to the topology τi.
Conversely, let x ∈ X. Then there exists τ1-open set Ux which meets only finitely
many of the sets (τi,τj)-g∗ωα-cl(Fλ), that is (τi,τj)-g∗ωα-cl(Fλ1 ), ..., (τi,τj)-
g∗ωα-cl(Fλn ). Let q ∈ Ux and q ∈ (τi,τj)-g∗ωα-cl(Fλk ) where k=1,2,...,n. Then
for each (τi,τj)-g∗ωα-open set Vq containing q such that Vq∩Fλk 6= φ. But q ∈ Ux
and so Ux meets only finitely many of the sets Fλ. Hence ξ is locally finite with
respect to the topology τi. 2

Lemma 5.1. Let f : (X,τ1,τ2) → (Y,σ1,σ2) be a function. Then f is (τi,τj)-
g∗ωα-closed if and only if for every y ∈ Y and U ∈ τ1O(X) which contains
f−1(y) there exists V ∈ (τi,τj)-g∗ωα-open set in (Y,σ1,σ2) such that y ∈ Y and
f−1(V ) ⊆ U.
Theorem 5.2. Let f : (X,τ1,τ2) → (Y,σ1,σ2) be (τi,τj)-g∗ωα-irresolute. If
ξ = {Fλ : λ ∈ Γ} be a (τi,τj)-P-locally finite collection in Y, then f−1(ξ) =
{f−1(Fλ) : λ ∈ Γ} is (τi,τj)-P locally finite collection in X.
Theorem 5.3. Let f : (X,τ1,τ2) → (Y,σ1,σ2) be (τi,τj)-g∗ωα-continuous. If
ξ = {Fλ : λ ∈ Γ} is (τi,τj)-P locally finite collection in Y, then f−1(ξ) =
{f−1(Fλ) : λ ∈ Γ} is locally finite collection with respect to the topology τi.
Definition 5.2. A non empty collection ξ = {Ai, i ∈ I, an index set} is called a
(τi,τj)-g∗ωα-open cover of a bitopological space (X,τ1,τ2) if X = ∪Ai and ξ ⊆
τ1-g∗ωαO(X,τ1,τ2) ∪ τ2-g∗ωαO(X,τ1,τ2) and ξ contains at least one member
of τ1-g∗ωαO(X,τ1,τ2) and one member of τ2-g∗ωαO(X,τ1,τ2).

Definition 5.3. A subset A of a bitopological space (X,τ1,τ2) is said to be (τi,τj)-
g∗ωα-compact if every cover of A by (τi,τj)-g∗ωα-open sets has a finite subcover.

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Example 5.1. Let X = {m,n,p,q}, τ1 = {X,φ,{m},{m,n}} and τ2 = {X,φ,
{m,n},{m,n,p},{m,p,q}}. Let ξ = {{m},{m,n},{m,n,p},{m,p,q}} be a
g∗ωα-open cover of (X,τ1,τ2). Then (X,τ1,τ2) is (τi,τj)-g∗ωα-compact.

Definition 5.4. A set A of a bitopological space (X,τ1,τ2) is said to be (τi,τj)-
g∗ωα-compact relative to X if every (τi,τj)-g∗ωα-open cover of A has a finite
subcover as a subspace.

Theorem 5.4. Every (τi,τj)-g∗ωα-compact space is (τi,τj) compact.
Proof: Let (X,τ1,τ2) be (τi,τj)-g∗ωα-compact. Let ξ = {Ai : i ∈ I} be (τi,τj)
open cover of X. Then X = ∪Ai and ξ ⊆ τi ∪ τj, so ξ contains at least one
member of τi and one member of τj. Since, every τi-open set is τi-g∗ωα-open,
we have X = ∪Ai and ξ ⊆ τi-g∗ωαO(X) ∪ τj-g∗ωαO(X) and by definition
ξ contains at least one member of τi-g∗ωαO(X,τ1,τ2) and one member of τj-
g∗ωαO(X,τ1,τ2). Therefore ξ is (τi,τj)-g∗ωα-open cover of X. As X is (τi,τj)-
g∗ωα-compact then ξ has the finite subcover and hence X is (τi,τj) compact.

Theorem 5.5. If Y is τi-g∗ωα closed subset of a (τi,τj)-g∗ωα-compact space
(X,τ1,τ2) then Y is τj-g∗ωα compact.
Proof: Let (X,τ1,τ2) be (τi,τj)-g∗ωα-compact. Let ξ = {Ai : i ∈ I} be a
τj-g∗ωα open cover of Y. As Y is τi-g∗ωα closed, Y c is τi-g∗ωα open. Also,
ξ ∪ Y c = Y c ∪ {Ai : i ∈ I} be a (τi,τj)-g∗ωα-open cover of X. Since X is
(τi,τj)-g∗ωα-compact, we have X = Y c ∪A1 ∪ ...∪An, so Y = A1 ∪ ....∪An.
Hence, Y is τj-g∗ωα compact.

Theorem 5.6. Let f : (X,τ1,τ2) → (Y,σ1,σ2) be a (τi,τj) continuous, bijective
and (τi,τj)-g∗ωα-irresolute. Then the image of a (τi,τj)-g∗ωα-compact space
under f is (τi,τj)-g∗ωα-compact.
Proof: Let f : (X,τ1,τ2) → (Y,σ1,σ2) be a (τi,τj) continuous surjective and
(τi,τj)-g∗ωα-closed. Let X be (τi,τj)-g∗ωα-compact. Let ξ = {Ai, i ∈ I} be
a (τi,τj)-g∗ωα-open cover of Y. Then Y = ∪Ai and ξ ⊆ σ1-g∗ωαO(Y ) ∪ σ2-
g∗ωαO(Y ) and ξ contains at least one member of σ1-g∗ωαO(Y ) and one member
of σ2-g∗ωαO(Y ). Therefore X = f−1(∪(Ai)) = ∪f−1(Ai) and f−1(ξ) ⊆ τ1-
g∗ωαO(X) ∪ τ2-g∗ωαO(X) and f−1(ξ) contains at least one member of τ1-
g∗ωαO(X) and one member of τ2-g∗ωαO(X). Therefore f−1(ξ) is the (τi,τj)-
g∗ωα-open cover of X. Since X is (τi,τj)-g∗ωα-compact, we have X = ∪f−1(Ai)
for each i = 1,...,n, that is Y = f(X) = ∪(Ai), i=1,...,n. Hence, ξ has the finite
subcover. Therefore Y is (τi,τj)-g∗ωα-compact.

Definition 5.5. A bitopological space (X,τ1,τ2) is said to be (τi,τj)-g∗ωα-paracompact
(pairwise g∗ωα paracompact) if every τi-open cover of X has a (τi,τj)-P-locally
finite (τi,τj)-g∗ωα-open refinement.

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Pairwise Paracompactness

Example 5.2. Let X = {m,n,p}, τ1 = {X,φ,{m},{m,n}} and τ2 = {X,φ,
{m},{p},{m,p}}. Let ξ = {{m},{m,n}}. Then the space (X,τ1,τ2) is (τi,τj)
g∗ωα-paracompact.

Definition 5.6. Let (X,τ1,τ2) be a bitopological space. Then X is said to be:
(a) (τi,τj)-g∗ωα-regular if for each τi-closed set F and x ∈ X there exist τi-
g∗ωα-open set U and τj-g∗ωα-open set V such that x ∈ U and F ⊆ V .
(b) (τi,τj)-g∗ωα-normal if there exist two disjoint τi-closed sets A and B , there
exist disjoint (τi,τj)-g∗ωα-open sets U and V such that A ⊆ U and B ⊆ V .

Example 5.3. Let X = {m,n,p}, τ1 = {X,φ,{m},{m,n}} and τ2 = {X,φ,
{m},{p},{m,p}}. Then (X,τ1,τ2) is (τi,τj)-g∗ωα-regular.

Example 5.4. Let X = {m,n,p}, τ1 = {X,φ,{m},{n,p}} and τ2 = {X,φ,
{m}}. Then (X,τ1,τ2) is (τi,τj)-g∗ωα-normal.

Proposition 5.1. A space (X,τ1,τ2) is (τi,τj)-g∗ωα-regular if and only if for
each τi-open set U and x ∈ U there exists V ∈ (τi,τj)-g∗ωαO(X) such that
x ∈ V ∈ (τi,τj)g∗ωα-cl(V ) ⊆ U.

Theorem 5.7. Every (τi,τj)-g∗ωα-paracompact pairwise Hausdorff bitopologi-
cal space is (τi,τj)-g∗ωα-regular.

Proof. Let A be τi-closed set with x /∈ A. Then for each y ∈ A, choose
τi-open sets Uy and Hx such that y ∈ Uy, x ∈ Hx and Uy ∩ Hx = φ, that is
x /∈ (τi,τj)g∗ωα-cl(Uy). Therefore the family U = {Uy : y ∈ A}∪{X − A}
is an τi-open cover of X and so it has a (τi,τj)-P-locally finite (τi,τj)-g∗ωα-open
refinement say ρ. Let V = {H ∈ ρ : H ∩A /∈ φ}, then V is a (τi,τj)-g∗ωα-open
set containing A and (τi,τj)-g∗ωα-cl(V ) = ∪{(τi,τj)-g∗ωα-cl(H) : H ∈ ρ and
H ∩A /∈ φ}. Therefore U = X − (τi,τj)-g∗ωα-cl(V ) is a (τi,τj)-g∗ωα-open set
containing x such that U and V are disjoint subsets of X. Thus X is (τi,τj)-g∗ωα-
regular. 2

Corolary 5.1. Every (τi,τj)-g∗ωα-paracompact pairwise Hausdorff bitopologcal
space (τi,τj)-g∗ωα-normal.

Theorem 5.8. Let (X,τ1) and (X,τ2) be two regular spaces. Then (X,τ1,τ2) is
(τi,τj)-g∗ωα-paracompact if and only if every τi-open cover ξ of X has a (τi,τj)-P
locally finite (τi,τj)-g∗ωα-closed refinement say ρ.

Proof. Necessity: Let ξ be an τ1-open cover of X. Then for each x ∈ X
choose a member Ux ∈ ξ. Since (X,τ1) and (X,τ2) are τi-regular, there exists
τi-open set Vx containing x such that τi ⊆ cl(Vx) ⊆ Ux. Therefore Ψ = {Vx :
x ∈ X} is an τi-open cover of X and by hypothesis Ψ has a (τi,τj)-P-locally

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P. S. Mirajakar and P. G. Patil

finite (τi,τj)-g∗ωα-refinement say Ω = {Wλ : λ ∈ Γ}. Consider the collection
(τi,τj)g

∗ωα− Ω = {(τi,τj)g∗ωα-cl(Wλ) : λ ∈ Γ} is a (τi,τj)-P-locally finite of
(τi,τj)-g∗ωα-closed subsets of (X,τ1,τ2). Since for every λ ∈ Γ, (τi,τj)-g∗ωα-
cl(Wλ) ⊆ (τi,τj)g∗ωα-cl(Vx) ⊆ τ1-cl(Vx) ⊆ Ux for some Ux ∈ ξ, therefore
(τi,τj)-g∗ωα-cl(Ω) is a refinement of ξ.
Sufficiency: Let ξ be an τi-open cover of X and Ψ be a (τi,τj)-P locally finite
(τi,τj)-g∗ωα-closed refinement of ξ. Then for each x ∈ X choose Wx ∈ (τi,τj)-
g∗ωαO(X) such that x ∈ Wx and Wx intersects at most finitely many member of
Ψ. Let Σ be (τi,τj)-g∗ωα-closed (τi,τj)-P-locally finite refinement of Ω = {Wx :
x ∈ X}. Then for each V ∈ Ψ, V 1 = X − H, where H ∈ Σ and H ∩ V = φ.
Then {V 1 : V ∈ Ψ} is a (τi,τj)-g∗ωα-open cover of X. Now for each V ∈ Ψ, let
us choose Uv ∈ Ξ such that V ⊆ Uv. Hence the collection {Uv ∩ V 1 : V ∈ Ψ}
is a (τi,τj)-P-locally finite (τi,τj)-g∗ωα-open refinement of ξ. Thus (X,τ1,τ2) is
(τi,τj)-g∗ωα-paracompact. 2

Theorem 5.9. Let A be a (i,j)regular closed subset of a bitopological space
(X,τ1,τ2). Then (A,τiA,τjA ) is (i,j)-g

∗ωα-paracompact.

Proof. Let Σ = {Vλ : λ ∈ Γ} is an τi-open cover of A in (A,τiA,τjA ).
Then for each λ ∈ Γ, choose an Uλ ∈ τi such that Vλ = A ∩ Uλ. Then the
collection ξ = {Uλ : λ ∈ Γ}∪{X −A} which is an τi-open cover of the (i,j)-
g∗ωα-paracompact space X and so it has a (i,j)-P-locally finite (i,j)-g∗ωα-open
refinement say Σ = {Wδ : δ ∈ ∆}. But we have (i,j)−RO(X) ⊆ (i,j)−O(X),
then the collection {A∩Wδ : δ ∈ ∆} is a (i,j)-P-locally finite (i,j)-g∗ωα-open
refinement of Σ in (A,τiA,τjA ). 2

Theorem 5.10. Let f : (X,τ1,τ2) → (Y,σ1,σ2) be (τ1,σ1) and (τ2,σ2) closed
(τi,τj)-g∗ωα-irresolute surjective function such that f−1(y) is τi-compact in (X,τ1)
for each y ∈ Y . If (Y,σ1,σ2) is (τi,τj)-g∗ωα-paracompact, then (X,τ1,τ2) is
also (τi,τj)-g∗ωα-paracompact.

Proof. Let ξ = {Uλ : λ ∈ Γ} be an τi-open cover of a bitopological space
(X,τ1,τ2). Then for each y ∈ Y , ξ is an τi-open cover of the τi-compact subspace
f−1(y). So there exist a finite subcover Γy of Γ such that f−1(y) ⊆ ∪Uλ for
each λ ∈ Ξλ. Let Uλ = ∪Uλ which is an τi-open in (X,τ1). As f is (τ1,σ1)-
closed, then for each y ∈ Y there exists σ1-open set Vy in Y such that y ∈ Vy
and f−1(Vy) ⊆ Uλ. Then the collection Ψ = {Vy : y ∈ Y} is an σ1-open
cover of the (τi,τj)-g∗ωα-paracompact space (Y,σ1,σ2) and so it has a (τi,τj)-
P locally finite (τi,τj)-g∗ωα-open refinement say Ω = {Wγ : γ ∈ ∆}. As f is
(τi,τj)-g∗ωα-irresolute, the collection f−1(Ω) = {f−1(Wγ) : γ ∈ ∆} which is an
(τi,τj)-g∗ωα-open (τi,τj)-P-locally finite cover of (X,τ1,τ2) such that for each
γ ∈ δ, f−1(Wγ) ⊆ Uy for some y ∈ Y . Then the collection {f−1(Wγ)∩Uγ : γ ∈

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Pairwise Paracompactness

δ,λ ∈ Γy} is an (τi,τj)-P-locally finite (τi,τj)-g∗ωα-open refinement of ξ. Thus
(X,τ1,τ2) is (τi,τj)-g∗ωα-paracompact. 2

Conclusion
The notions of sets and functions in topological spaces are extensively devel-

oped and used in many fields such as particle physics, computational topology,
quantum physics. By researching generalizations of closed sets, some new Para-
compact spaces have been founded and they turn out to be useful in the study of
digital topology.

6 Acknowledgment
The second author wish to thanks to University Grant Commission, New Delhi

for financial support.

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