INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,  
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)  
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIV, Issue XI, November 2025  
A Study on 흁̂휷 Connectedness in Bitopological Spaces  
Dr. J. Subashini  
Department of Mathematics, Sri Ramakrishna College of Arts and Science for Women, New  
Siddhapudur, Coimbatore - 641 044  
DOI : https://doi.org/10.51583/IJLTEMAS.2025.1411000086  
Received: 01 December 2025; Accepted: 06 December 2025; Published: 18 December 2025  
ABSTRACT  
The objective of this paper is to study a special case of connectedness in bitopological spaces by considering  
휏1휏2 휇훽 open sets and examining their relationships with 휏1휏2 connected space and 휏1휏2 pre-connected space.  
̂
Key words: 휏1휏2 휇̂훽 open set , 휏1휏2휇̂훽 connected space .  
INTRODUCTION  
The study of bitopological spaces was initiated by Kelly, J .C [5] . A triple ( X, 휏1, 휏2 ) is called bitopological  
space if (X , 휏1 ) and ( X , 휏1 ) are two topological spaces . In 1997, Kumar Sampath S [6] introduced the  
concept of 휏1휏2 - -open sets in bitopological spaces. In 1981, Bose S [1] introduced the notion of 휏1휏2 - semi  
- open sets in bitopological spaces . In 1992, Kar A [4] have introduced the notion of 휏1휏2 - pre- open sets in  
bitopological spaces . In 2012 , H . I Al-Rubaye Qaye [2] introduced the notion of 휏1휏2 - semi - -open sets  
in bitopological spaces . In this paper, we study a special case of 휇훽 connectedness in bitopological spaces, and  
̂
we prove several results by comparing them with similar cases in topological spaces.  
Preliminaries  
Throughout the paper, spaces always mean a bitopological spaces , the closure and the interior of any subset A  
of X with respect to 휏푖 , will be denoted by 휏푖cl A , and 휏푖int A respectively, for  
i 1,2.  
Definition 2.1 :  
Let (X, 휏1, 휏2 ) be a bitopological space , A  
(i) 휏1휏2 pre-open set [4] if A 휏1int(휏2cl(A)).  
(ii) 휏1휏2휇훽 closed set [1] if 휏2 휇푐푙(퐴) U , whenever A  
(iii) 휏1휏2 open set [6] if A 휏1int(휏2cl(휏1int(A))).  
Remark 2.2 :  
The family of 휏1휏2pre-open ( resp. 휏1휏2휇  
X , A is said to be :  
̂
̂
U, U is 훽 open in 휏1.  
̂
훽 open ) sets of X is denoted by  
휏1휏2 PO(X) ( resp. 휏1휏2 휇  
̂
훽O(X) ).  
Example 2.3 :  
Let X {a, b, c, d},  
a bitopological space . The family of all 휏1휏2휇  
c}, {a, d}, {b, c}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, X}.  
X,  
a},{b},{a, b},{a, b, c}} , and  
훽 open sets of X is : 휏1휏2 휇  
X,  
a},{c},{a,c},{a,b,c} is  
̂
̂
훽O(X) = {∅, {a}, {b}, {c}, {a, b}, {a,  
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Remark 2.4 :  
It is clear by definition that in any bitopological space the following hold :  
(i) Every 휏1 open set is 휏1휏2 휇훽 open , 휏1휏2semi -open , 휏1휏2 open set .  
(ii) Every 휏1휏2 -open set is 휏1휏2pre-open , 휏1휏2semi open set .  
̂
(iii) The concept of 휏1휏2pre-open and 휏1휏2semi open sets are independent .  
Proposition 2.5 :  
A subset A of a bitopological space (X, 휏1, 휏2) is 휏1휏2  
open set if and only if there exists  
an 휏1 open set U , such that U  
A
휏1int(휏2cl(U)) .  
Proof :  
This follows directly from the definition (2.1) (iii).  
Proposition 2.6 : [7]  
A subset A of a bitopological space (X, 휏1, 휏2) is 휏1휏2semi open set if and only if there  
exists an 휏1open set  
U, such that U  
A
휏2cl(U).  
Proposition 2.7 : [3]  
A subset A of a bitopological space (X, 휏1, 휏2) is 휏1휏2pre open set if and only if there exists  
an 휏1 open set U , such that A U  
휏2cl(A).  
Theorem 2.8 :  
A subset A of a bitopological space (X, 휏1, 휏2) is an 휏1휏2  
open set if and only if A is 휏1휏2semi open set and  
휏1휏2 pre open set .  
Proof :  
Follows from definition (2.1) and remark (2.5) .  
Definition 2.9 : [4,6]  
Let (X, 휏1, 휏2)  
be a bitopological space and A  
X ,the intersection of all 휏1휏2  
closed  
(resp . 휏1휏2pre closed ) sets containing A is called 휏1휏2  
closure (resp . 휏1휏2pre closure )of A , and is denoted  
by 휏1휏2 cl(A) (resp . 휏1휏2 pcl(A) ) ;  
i.e 휏1휏2 cl(A)  
{B  
X : B is 휏1휏2  
closed set , A  
X} .  
X} and  
휏1휏2pcl(A)  
{B  
X : B is 휏1휏2pre closed set , A  
Remark 2.10 : [2]  
(i) Every 휏1 open set is 휏1휏2휇  
̂
훽 -open set, but the converse need not be true.  
(ii) If every 휏1 open set is 휏1 closed and every nowhere 휏1 dense set  
is 휏1 closed  
in any  
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bitopological space , then every 휏1휏2휇  
̂
훽 open set is an 휏1 open set.  
Remark 2.11 : [2]  
(i) Every 휏1휏2  
open set is 휏1휏2휇  
̂
훽 open set, but the converse is not true in general .  
(ii) If every  
휏1 open set  
is 휏1 closed  
set in any bitopological space, then every 휏1휏2휇̂훽 open set is an  
휏1휏2  
open set .  
Remark 2.12 : [2]  
The concepts of 휏1휏2휇  
̂
훽-open and 휏1휏2 pre open sets are independent , as the following  
example.  
Example 2.13:  
In example (2.3) , {b,c}is a 휏1휏2- pre-open set but not 휏1휏2 - semi -  
-open set .  
Remark 2.14 : [2]  
(i) It is clear that every 휏1휏2 semi open and 휏1휏2 pre open subsets of any bitopological space is  
휏1휏2휇훽 open set  
(ii) An 휏1휏2휇훽 open set in any bitopological space (X, 휏1, 휏2) is 휏1휏2 pre open set if every 휏1 open subset of  
̂
̂
X is 휏1 closed set ( from remark (2.17) (ii) and remark (2.5) (iii) ) .  
Definition 2.15 : [2]  
The complement of 휏1휏2휇  
̂
훽 open set is called 휏1휏2휇  
̂
훽 closed set . Then family of all 휏1휏2휇̂훽 closed sets of X  
( )  
is denoted by 휏1휏2휇  
̂
훽퐶 푋 .  
Definition 2.16 : [2]  
Let (X, 휏1, 휏2) be a bitopological space and A  
X ,the intersection of all 휏1휏2휇  
̂
훽 closed sets containing A is  
called 휏1휏2휇훽 closure of A , and is denoted by 휏1휏2휇  
̂
̂
훽푐푙(퐴) ;  
i.e 휏1휏2휇  
̂
훽cl(A)  
{B  
X : B is 휏1휏2휇  
̂
훽 closed set , A  
X}.  
Remark 2.17 : [2]  
The following diagram shows the relations among the different types of weakly open sets  
that were studied in this section:  
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̂
흉ퟏ흉ퟐ흁휷 Connectedness in Bitopological Spaces :  
In this section the notion of 휏1휏2휇  
with 휏1휏2 connected space and 휏1휏2 pre connected space are studied.  
̂
훽 connected space is introduced in bitopological spaces and their relationships  
Definition 3.1 :  
Let (X, 휏1, 휏2)  
be a bitopological space , two non-empty subsets A and B of X are said to be 휏1휏2휇  
̂
훽 separated  
if A  
휏1휏2휇  
̂
훽cl(B) and 휏1휏2휇훽cl(A)  
̂
B
.
Definition 3.2 :  
A bitopological space (X, 휏1, 휏2) is called 휏1휏2휇  
̂
훽 connected if it is not the union of two non empty 휏1휏2휇  
̂
훽
separated 휏1휏2휇훽 opensets .  
̂
A subset B  
An 휏1휏2휇훽 disconnection of X is a pair of complement, non-empty, 휏1휏2휇  
subsets.  
Remark 3.3 :  
X is 휏1휏2휇  
̂
훽 connected if it is 휏1휏2휇  
̂
훽 connected as a subspace of X.  
̂
̂
훽 open 휏1휏2휇  
̂
훽 closed  
The only 휏1휏2 휇  
̂
훽 open 휏1휏2휇  
̂
훽 closed subsets in 휏1휏2휇  
̂
훽 connected space X are X and  
.
Remark 3.4 :  
Every 휏1휏2휇  
Proof :  
Suppose that X is not 휏1 connected, then there exist an two non-empty A , B are 휏1 open such that  
̂
훽 connected space is 휏1 connected, but the converse is not true.  
A
B
and A  
B
X . we have A , B are 휏1휏2휇  
̂
훽 open sets , A  
B
X and  
A
B
, hence X is not 휏1휏2휇훽 connected which is a contradiction. Thus, X is 휏1 connected.  
̂
But the converse is not true as in the following example.  
Example 3.5 :  
Let X {a, b, c, d},  
a bitopological space . The family of all 휏1휏2휇  
c}, {a, d}, {b, c}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, X}.  
X,  
a},{b},{a, b},{a, b, c}} , and  
X,  
a},{c},{a,c},{a,b,c} is  
̂
훽 open sets of X is : 휏1휏2 휇  
̂
훽O(X) = {∅, {a}, {b}, {c}, {a, b}, {a,  
Then X is  
connected space, but X is not 휏1휏2휇̂훽 connected.  
Remark 3.6 :  
If every 휏1 open set  
is 휏1 closed and every nowhere 휏1 dense set  
is 휏1 closed  
in any  
bitopological space , then every 휏1 connected space is 휏1휏2휇  
̂
훽 connected .  
Proof :  
Follows from remark ( 2.16 ) (ii) .  
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Definition 3.7 :  
A function f : (X, 휏1, 휏2)  
f (U) is 휏1휏2휇훽 open in Y.  
(X, 휎1, 휎2)  
is called 휏1휏2휇̂훽 open if for each 휏1 open set U of X ,  
̂
Definition 3.8 :  
A function f : (X, 휏1, 휏2)  
휏1 open subset of Y is 휏1휏2휇  
(X, 휎1, 휎2) is called 휏1휏2휇  
̂훽 open subset of X.  
̂
훽 continuous if and only if the inverse image of each  
Proposition 3.9 :  
Every 휏1 continuous function is 휏1휏2휇훽 continuous.  
̂
Proof :  
Follows from remark ( 2.16 ) (i) .  
Definition 3.10 :  
A function f : (X, 휏1, 휏2)  
(X, 휎1, 휎2)  
is called 휏1휏2휇̂훽 irresolute if and only if the inverse image of each  
휏1휏2휇훽 open subset of Y is 휏1휏2휇  
̂
̂
훽 open subset of X .  
Proposition 3.11 :  
Every 휏1휏2휇훽 irresolute function is 휏1휏2휇훽 continuous.  
̂
̂
Proof :  
Let A be any 휎1 open set  
in Y. Then we have A is an open set in 휏1휏2휇  
̂
훽 open in Y [ from remark ( 2.16) (i) ] .  
Since f is 휏1휏2휇  
̂
훽 irresolute function, then 푓−1(A) is 휏1휏2휇  
̂
훽 open set in X . Therefore f is 휏1휏2휇훽 continuous.  
̂
Proposition 3.12 :  
Let f : (X, 휏1, 휏2)  
(X, 휎1, 휎2)  
be two bitopological spaces . If X is 휏1휏2휇̂훽 connected and f is 휏1휏2휇̂훽  
continuous function from (X, 휏1, 휏2) onto (X, 휎1, 휎2) , then Y is 휏1 connected.  
Proof :  
Suppose that A is an 휏1 open 휏1 closed subset of Y, then 푓−1(A) is 휏1휏2휇  
훽 open 휏1휏2휇̂훽 closed in X . Hence  
̂
푓−1(A)  
is  
or X , but X is 휏1휏2휇  
̂
훽 connected . So A is  
or Y. Hence Y is 휏1 connected.  
Proposition 3.13 :  
An 휏1휏2휇훽 irresolute image of any 휏1휏2휇  
Proof :  
̂
̂훽 connected bitopological space is 휏1휏2휇̂훽 connected.  
Follows directly from proposition ( 3.12 ) .  
Definition 3.14 :  
Let (X, 휏1, 휏2) be a bitopological space , two non-empty subsets A and B of X are said to  
be 휏1휏2 separated ( resp. 휏1휏2 pre separated ) if A 휏1휏2 cl(B) ( resp. A 휏1휏2푝cl(B)  
)
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and 휏1휏2 cl(A)  
B
(resp . 휏1휏2 pcl(A)  
B
) .  
Definition 3.15 :  
A bitopological space (X, 휏1, 휏2) is called 휏1휏2  
if it is not the union of two non-empty 휏1휏2  
( resp. 휏1휏2 pre open ) sets.  
connected ( resp. 휏1휏2 pre connected ) space  
separated ( resp. 휏1휏2 pre separated ) 휏1휏2 open  
Proposition 3.16 :  
Every 휏1휏2 pre connected space is 휏1휏2  
Proof :  
connected.  
Follows from remark (2.5) (ii).  
Proposition 3.17 :  
If every 휏1휏2 pre open set in a bitopological space (X, 휏1, 휏2) is 휏1휏2 semi open set , then X is  
휏1휏2 pre connected space , whenever it is an 휏1휏2  
Proof :  
connected space .  
Follows from theorem (2.9).  
Proposition 3.18 :  
Every 휏1휏2 pre connected space is 휏1 connected.  
Proof :  
Follows from remark (2.5) (i).  
Remark 3.19 :  
Every 휏1휏2휇  
̂
훽 connected space is 휏1휏2  
connected.  
Proof :  
Follows from remark (2.17) (i).  
Proposition 3.20 :  
In a bitopological space (X, 휏1, 휏2) if every 휏1 open subset  
of X is 휏1 closed  
set , then X is 휏1휏2휇̂훽  
connected space , whenever it is an 휏1휏2  
Proof :Follows from remark (2.17) (ii).  
Remark 3.21 :  
connected space.  
The concepts of 휏1휏2 pre connected space and 휏1휏2휇훽 connected space are independent.  
̂
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Proposition 3.22 :  
If every 휏1 open set  
whenever it is 휏1휏2 pre connected.  
in a bitopological space (X, 휏1, 휏2) is 휏1 closed , then X is 휏1휏2휇̂훽 connected ,  
Proof :  
Follows from propositions (3.16) and (3.20).  
Proposition 3.23 :  
If every 휏1휏2 pre open set in a bitopological space (X, 휏1, 휏2) is 휏1휏2 semi open set , then X is  
휏1휏2 pre connected space, whenever it is an 휏1휏2휇훽 connected space.  
̂
Proof :  
Follows from remark (3.19) and proposition (3.20).  
Remark 3.24 :  
The following diagram shows the relations among the different types of connectedness:  
REFERENCES  
1. Bose, S., “Semi-open sets, semi-continuity and semi-open mappings in bitopological spaces,” Bull. Cal.  
Math. Soc., 73 (1981), 237–246.  
2. Al-Rubaye, H. I., Qaye, “Semi-α-separation axioms in bitopological spaces,” Al-Muthanna Journal of  
Pure Sciences, 1(1) (2012), 190–206.  
3. Jelic, M., “A decomposition of pairwise continuity,” J. Inst. Math. Comput. Sci. Math. Ser., 3 (1990),  
25–29.  
4. Kar, A., Bhattacharyya, P., “Bitopological pre-open sets, pre-continuity and pre-open mappings,” Indian  
J. Math., 34 (1992), 295–309.  
5. Kelly, J. C., “Bitopological spaces,” Proc. London Math. Soc., 13 (1963), 17–89.  
6. Kumar Sampath, S., “On a decomposition of pairwise continuity,” Bull. Cal. Math. Soc., 89 (1997), 441–  
446.  
7. Maheshwari, S. N., Prasad, R., “Semi-open sets and semi-continuous functions in bitopological spaces,”  
Math. Notae, 26 (1977/78), 29–37.  
8. J.Subashini and K.Indirani, “On μ̂β set and continuity in Topological Spaces” (proceeding)  
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