Ratio Mathematica                                                                            Volume 45, 2023 

 

 
 

 

 

 

 

Tri- 𝒃𝒈 ̂Closed sets in Tri- Topological Spaces 
 
 

L. Jeyasudha* 

K. Bala Deepa Arasi† 
 

 

 

Abstract 

In this paper, we introduce a new class of sets called tri- 𝑏𝑔 ̂closed sets and tri- 𝑏𝑔 ̂open 
sets via the concept of tri- 𝑔 ̂closed sets in tri topological spaces. Also, we investigate 
the relationship with other existing closed sets in tri-topological space.  

Keywords: Tri- 𝑏𝑔 ̂closed sets, tri- 𝑏𝑔 ̂open sets, tri- 𝑏𝑔 ̂closure, tri- 𝑏𝑔 ̂interior. 
 

2010 AMS subject classification:  54A40‡. 

 
 
 

  

 
*Research Scholar, Reg. No: 20122012092004, PG & Research Department of Mathematics, A.P.C. 

Mahalaxmi College for Women, Thoothukudi, TN, India. Affiliated to Manonmaniam Sundaranar 

University, Tirunelveli, TN, India. E. mail: jeyasudha555@gmail.com.  
†Assistant Professor of Mathematics, A.P.C. Mahalaxmi College for Women, Thoothukudi, Tamilnadu, 

India. E. mail: baladeepa85@gmail.com 
‡ Received on July 18, 2022. Accepted on October 15, 2022. Published on January 30, 2023. doi: 

10.23755/rm.v45i0.1002. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published 

under the CC-BY license agreement. 

142

mailto:jeyasudha555@gmail.com
mailto:baladeepa85@gmail.com


L. Jeyasudha, K. Bala Deepa Arasi 

 

1. Introduction 

 The concept of tri- topological space was first initiated by M. Kovar [6] in 2000, in 

2003, R. Subasree and M. Maria Singam [10] defined 𝑏𝑔 ̂- closed sets in topological 
spaces. In [3], we introduced tri- ĝ closed sets in tri- topological spaces and studied their 
properties. In this paper, we define tri- 𝑏𝑔 ̂closed sets and tri- 𝑏𝑔 ̂open sets via the 
concept of tri- 𝑔 ̂closed sets. Also, we investigate the relationship with other existing 
closed sets in tri- topological space. 

 

2. Preliminaries 

Throughout this paper (X, τ1, τ2, τ3) (or simply X) represents tri- topological spaces 

on which no separation axioms are assumed unless other wised mentioned. For a subset 

A of (X,τ1,τ2,τ3), tri- cl(A), tri- int(A) and A
c denote the tri- closure of A, tri- interior of 

A and compliment of A respectively.  

Definition 2.1 Let X be a non-empty set. A family τ of subsets of X is said to be a 

topology on X, if τ satisfies the following axioms. 

a) ф, X ∈τ,  

b) If Ai∈τ for i = 1,2,…..,n, then ⋂ 𝐴
𝑛
𝑖=1 i ∈τ,  

c) If Aα∈τ for α ∈ I, then ⋃ 𝐴𝛼 α∈ τ. 

The pair (X, τ) is called a topological space and any set A in 𝜏 is called an open set. The 
complement of an open set A is called closed set. 

Definition 2.2 Let X be a non-empty set. A family G of subsets of X is said to be a 

generalized topology on X, if G satisfies the followings axioms. 

a) ф ∈ G,   

b) If Aα∈G for α ∈I, then ⋃ 𝐴𝛼 α∈ G.  

The pair (X, G) is called a generalized topological space.  

Definition 2.3 Let X be a non-empty set. A family τ* of subsets of X is said to be a 

Supra topology on X, if τ* satisfies the following axioms. 

a) ф, X ∈ τ*,  

b) If Aα∈ τ* for α ∈ I, then ⋃ 𝐴𝛼 α∈ τ*.  

The pair (X, τ*) is called a Supra topological space. 

Definition 2.4 Let X be a non-empty set. A family τiX of subsets of X is said to be a 

Infra topology on X, if τiX satisfies the following axioms. 

a) ф, X ∈ τiX,  

b) If Ai∈ τiX for i = 1, 2… n, then ⋂ 𝐴
𝑛
𝑖=1 i ∈ τiX.  

The pair (X, τiX) is called Infra topological space.  

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Tri – gb ˆ  Closed Sets in Tri- Topological Spaces 

 

 

 

Definition 2.5 Let (X, τ) be a topological space then τ is said to be indiscrete topology 

if τ is a collection of only X and ф. Indiscrete topology is also known as trivial 

topology. 

Definition 2.6 Let (X, τ) be a topological space then τ is said to be discrete topology if τ 

is a collection of all subsets of X. 

Definition 2.7 Let (X, τ) be a topological space then a subset A of X is said to be 𝑏𝑔 ̂-

closed set if bcl (A) ⊆ U whenever A ⊆ U, U is ĝ- open in X.  

Definition 2.8 Let X be a nonempty set and τ1, τ2 and τ3 are topologies on X. Then a 

subset A of X is said to be tri- open set if A ∈ τ1∪τ2∪τ3 and its complement is said to 

be tri- closed set and X with three topologies called tri- topological spaces (X, τ1, τ2, τ3). 

Definition 2.9 Let (X, τ1, τ2, τ3) be a tri- topological space and let A ⊆ X. The union of 

all tri- open sets contained in A is called the tri- interior of A. The intersection of all tri- 

closed sets containing A is called the tri- closure of A. 

Definition 2.10 Let (X, τ1, τ2, τ3) be a tri- topological space. A ⊆ X is said to be  

1) A tri- α open set if A ⊆ tri- int (tri- cl (tri- int (A))). 

2) A tri- b open set if A ⊆ [tri- cl (tri- int (A))] ∪ [tri- int (tri- cl (A))]. 

3) A tri- semi closed set if tri- int (tri- cl (A)) ⊆ A. 

4) A tri- g closed set if tri- cl (A) ⊆ U whenever A ⊆ U and U is tri- open set in X. 

5) A tri- gs closed set if tri- scl (A) ⊆ U whenever A ⊆ U and U is tri- open set in X. 

6) A tri- bτ closed set if tri- clb (A) ⊆ U whenever A ⊆ U and U is tri- open set in X. 

7) A tri- g*bw closed set if tri- bcl (A) ⊆ U whenever A ⊆ U, U is tri- gs open in X. 

8) A tri- ĝ closed set if tri- cl (A) ⊆ U whenever A ⊆U, U is tri- semi open in X. 

The complement of tri- α open set, tri- b open set, tri- semi closed set, tri- g closed set, 

tri- gs closed set, tri- bτ closed set, tri- g*bw closed set and tri- ĝ closed setis called tri- 
α closed set, tri- b closed set, tri- semi open set, tri- g open set, tri- gs open set, tri- bτ 

open set, tri- g*bw open set and tri- ĝ open set respectively. 

Theorems 2.11 

1) Every tri- closed set is tri- semi closed. 
2) Every tri- closed set is tri- b closed. 
3) Every tri- closed set is tri- gs closed. 
4) Every tri- closed set is tri- bτ closed. 
5) Every tri- closed set is tri- g*bω closed. 
6) Every tri- closed set is tri- ĝclosed set. 
7) Every tri- semi closed set is tri- gs closed. 

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8) Every tri- semi closed set is tri- b closed. 
9) Every tri- semi closed set is tri- g*bω closed. 
10) Every tri- b closed set is tri- bτ closed. 
11) Every tri- semi closed set is tri- bτ closed. 
12) Every tri- α closed set is tri- b closed set.  
13) Every tri- g*bω closed set is tri- bτ closed. 
14) Every tri- ĝ closed set is tri- g closed. 
15) Every tri- ĝ closed set is tri- gs closed. 

 

3. Tri- 𝑏𝑔 ̂Closed Sets in Tri- Topological Space  
We introduce the following definitions 

Definition 3.1 Let (X,τ1,τ2,τ3) be a tri- topological space then a subset A of X is said to 

be tri- 𝑏𝑔 ̂closed set if tri- bcl (A) ⊆ U whenever A ⊆ U, U is tri- ĝ open in X. The 

family of all tri- 𝑏𝑔 ̂closed sets of X is denoted by tri- 𝑏𝑔 ̂ C(X). 
 

Example 3.2 Let X = {a, b, c} with the topologies τ1 = {X, ф, {a, b}}, τ2 = {X, ф, {b, 

c}}, τ3 = {X, ф, {a, c}}, Open sets in tri- topological spaces are union of all three 

topologies.   τ1 ∪ τ2 ∪ τ3 = {X, ф, {a, b}, {b, c}, {a, c}}; Tri- ĝO(X) = {X, ф, {a, b}, {b, 

c}, {a, c}}; Hence tri- 𝑏𝑔 ̂C(X) = {X, ф, {a}, {b}, {c}}. 

Remark 3.3 ф and X are always tri- 𝑏𝑔 ̂closed set. 
 

Remark 3.4 Intersection of tri- 𝑏𝑔 ̂closed sets need not be tri- 𝑏𝑔 ̂closed set. 
 

Example 3.5 Let X = {a, b, c}, τ1 = {X, ф}, τ2 = τ3 = {X, ф, {a}}, tri- gb ˆ C(X) = {X, ф, 

{b}, {c}, {a, b}, {b, c}, {a, c}}. Here, {a, b}, {a, c} are tri- 𝑏𝑔 ̂closed sets but {a, b} ∩ 
{a, c} = {a} is not a tri- 𝑏𝑔 ̂closed set. 

Remark 3.6 Union of tri- 𝑏𝑔 ̂closed sets need not be tri- 𝑏𝑔 ̂ closed set. 
 

Example 3.7 Let X = {a, b, c}, τ1 = {X, ф, {a, c}}, τ2 = {X, ф, {b}, {b, c}}, τ3 = {X, ф, 

{c}, {a, b}}, tri- 𝑏𝑔 ̂C(X) = {X, ф, {a}, {b}, {c}, {a, b}, {a, c}}. Here, {b}, {c} are tri-

 𝑏𝑔 ̂closed sets but {b} ∪ {c} = {b, c} ∉ tri- 𝑏𝑔 ̂C(X). 

Remark 3.8 Difference of two tri- 𝑏𝑔 ̂closed sets need not be tri- 𝑏𝑔 ̂closed set. 
 

Example 3.9 In previous example – 3.7, tri- 𝑏𝑔 ̂C(X) = {X, ф, {a}, {b}, {c}, {a, b}, {a, 
c}}. Let A = X and B = {a}, Also A and B are tri- 𝑏𝑔 ̂closed sets. But A \ B = X \ {a} = 
{b, c} is not a tri- 𝑏𝑔 ̂closed set. 

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Tri – gb ˆ  Closed Sets in Tri- Topological Spaces 

 

 

 

Remark 3.10 

1) (X, Tri- 𝑏𝑔 ̂C(X)) need not be Topological space. 
2) (X, Tri- 𝑏𝑔 ̂C(X)) need not be Generalized topological space. 
3) (X, Tri- 𝑏𝑔 ̂C(X)) need not be Supra topological space. 
4) (X, Tri- 𝑏𝑔 ̂C(X)) need not be Infra topological space. 
 

Example 3.11 In examples – 3.5, 3.7 we get the results. 

Definition 3.12 Let (X, τ1, τ2, τ3) be a tri- topological space. The intersection of all tri- 

𝑏𝑔 ̂closed sets of X containing a subset A of X is called tri- 𝑏𝑔 ̂closure of A and is 

denoted by tri- 𝑏𝑔 ̂cl(A). (i.e) tri- 𝑏𝑔 ̂cl (A) = ∩ {B ⊆ X: B ⊇ A and B is tri- 𝑏𝑔 ̂ closed 

set}. 

Remark 3.13   

1)   tri- 𝑏𝑔 ̂cl(ф) = ф,    
2)   tri- 𝑏𝑔 ̂cl(X) = X,    

3)   A ⊆ tri- 𝑏𝑔 ̂cl(A),    

4)   tri- 𝑏𝑔 ̂cl(A) = tri- 𝑏𝑔 ̂cl(tri- 𝑏𝑔 ̂cl(A)).  

Proposition 3.14 Let (X,τ1,τ2,τ3) be a tri- topological space. Let A ⊆ X, Then A = tri-

 𝑏𝑔 ̂ cl (A) if A is tri- 𝑏𝑔 ̂closed set. 

Proof. Suppose A is a tri- 𝑏𝑔 ̂closed set in X then, tri- bcl (A) ⊆ U whenever A⊆U, U 

is tri- ĝ open in X. Since, A⊇A and A is tri- 𝑏𝑔 ̂closed set. Let B ⊆ X then A ∈{B ⊆ X 

: B ⊇ A and B is tri- 𝑏𝑔 ̂closed} ⇒ A = ∩ {B ⊆ X : B ⊇ A and B is tri- 𝑏𝑔 ̂closed}. 

Hence A= tri- 𝑏𝑔 ̂cl(A). 

Remark 3.15 The tri- 𝑏𝑔 ̂closure of a set A is not always tri- 𝑏𝑔 ̂closed set. 
 

Example 3.16 Let X = {a, b, c}, τ1 = {X, ф}, τ2 = τ3 = {X, ф,{a}}, tri- 𝑏𝑔 ̂C(X) = {X, ф, 
{b},{c},{a, b},{b, c}, {a, c}}.Here, tri- 𝑏𝑔 ̂cl({a}) = {a} is not a tri- 𝑏𝑔 ̂closed set. 

Proposition 3.17 Every tri- b closed set is tri- 𝑏𝑔 ̂closed set. 
Proof: Let A be any tri- b closed set in X and U be any tri- ĝ open set in X such that      

A⊆U. Since, A is tri- b closed then tri- bcl (A) =A for every subset A of X. tri- bcl(A) = 

A ⊆ U. Hence A is tri- 𝑏𝑔 ̂closed set. 

Converse of the above proposition need not be true as seen in the following example. 

 

Example 3.18 Let X = {a, b, c}, τ1 = {X, ф}, τ2 = τ3 = {X, ф,{a}}, tri- b C(X) = {X, 

ф,{b}, {c},{b, c}; tri- 𝑏𝑔 ̂C(X) = {X, ф, {b}, {c}, {a, b}, {b, c}, {a, c}}; here {a, b}, {a, 
c} are tri- 𝑏𝑔 ̂ closed sets but not a tri- b closed set. 

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Proposition 3.19 Every tri- closed set is tri- 𝑏𝑔 ̂closed set. 
Proof: Let A be any tri- closed set in X. Since every tri- closed set is tri- b closed set. 

Therefore, A is tri- b closed set in X. By proposition 3.17, A is tri- 𝑏𝑔 ̂closed set. 
Converse of the above proposition need not be true as seen in the following example. 

 

Example 3.20 Let X = {a, b, c}, τ1 = {X, ф, {a}}, τ2 = {X, ф, {b}}, τ3 = {X, ф, {a, c}},       

tri- C(X) = {X, ф, {b}, {a, c}, {b, c}}; tri- 𝑏𝑔 ̂C(X) = {X, ф, {b}, {c}, {b, c}, {a, c}}; 
here {c} is tri- 𝑏𝑔 ̂closed set but not a tri- closed set. 

Proposition 3.21 Every tri- semi closed set is tri- 𝑏𝑔 ̂closed set. 
Proof: Let A be any tri- semi closed set in X. Since every tri- semi closed set is tri- b 

closed set. Therefore, A is tri- b closed set in X. By proposition 3.17, A is tri- 𝑏𝑔 ̂closed 
Converse of the above proposition need not be true as seen in the following example. 

 

Example 3.22 Let X = {a, b, c}, τ1 = {X, ф, {a}}, τ2  = {X, ф, {a, b}}, τ3 = {X, ф, {b, 

c}}, tri- sC(X) = {X, ф, {a}, {c}, {b, c}}; tri- 𝑏𝑔 ̂C(X) = {X, ф, {a}, {b}, {c}, {b, c}, 
{a, c}}; here {b},{a, c} are tri- 𝑏𝑔 ̂closed sets but not a tri-semi closed set. 

Proposition 3.23 Every tri- α closed set is tri- 𝑏𝑔 ̂closed set. 
Proof: Let A be any tri- α closed set in X. Since every tri- α closed set is tri- b closed 

set. Therefore, A is tri- b closed set in X. By proposition 3.17, A is tri- 𝑏𝑔 ̂closed set. 
Converse of the above proposition need not be true as seen in the following example. 

 

Example 3.24 Let X = {a, b, c}, τ1 = τ2 = {X, ф, {a}}, τ3 = {X, ф, {b, c}}, tri- α C(X) = 

{X, ф, {a}, {b, c}}; tri- 𝑏𝑔 ̂C(X) = {X, ф, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}}; here 
{b}, {c}, {a, b}, {b, c} are tri- 𝑏𝑔 ̂closed sets but not a tri-α closed set. 

Proposition 3.25 Every tri-g*bω closed set is tri- 𝑏𝑔 ̂closed set. 

Proof: Let A be any tri-g*bω closed set in X and A ⊆ U, where U is tri- ĝ open set in 

X. Since, every tri- ĝ open set is tri- gs open. Therefore, U is tri- gs open in X. Since, A 

is tri- g*bω closed set in X then tri- bcl(A) ⊆ U. Hence A is tri- 𝑏𝑔 ̂closed set in X. 

Converse of the above proposition need not be true as seen in the following example. 

 

Example 3.26 Let X = {a, b, c}, τ1 = {X, ф}, τ2 = {X, ф, {a}}, τ3 = {X, ф, {a, b}}, tri-

g*bω C(X) = {X, ф, {b}, {c}, {b, c}}; tri- 𝑏𝑔 ̂C(X) = {X, ф, {b}, {c}, {b, c}, {a, c}}; 
here {a, c} is tri- 𝑏𝑔 ̂closed set but not a tri- g*bω closed set. 

Proposition 3.27 Every tri- 𝑏𝑔 ̂closed set is tri- bτ closed set. 

Proof: Let A be any tri- 𝑏𝑔 ̂closed set in X and A ⊆ U, where U is tri- open set in X. 

Since, every tri- open set is tri- ĝ open. Therefore, U is tri- ĝ open in X. Since, A is tri- 

𝑏𝑔 ̂closed set in X then tri- bcl(A) ⊆ U. Hence A is tri- bτ closed set in X. 

Converse of the above proposition need not be true as seen in the following example. 

147



Tri – gb ˆ  Closed Sets in Tri- Topological Spaces 

 

 

 

Example 3.28 Let X = {a, b, c}, τ1 = {X, ф, {a}}, τ2 = {X, ф, {b}}, τ3 = {X, ф, {a, c}},       

tri- 𝑏𝑔 ̂C(X) = {X, ф, {b}, {c}, {b, c}, {a, c}}; tri- bτ C(X) = {X, ф, {b}, {c}, {a, b}, 
{b, c}, {a, c}}; here {a, b} is tri- bτ closed set but not a tri- 𝑏𝑔 ̂closed set. 

Remark 3.29 Tri- g closed sets and tri- 𝑏𝑔 ̂closed sets are independent. 
 

Example 3.30 Let X = {a, b, c}, τ1 = {X, ф, {a}}, τ2 = {X, ф, {b}}, τ3 = {X, ф, {a, b}},       

tri- gC(X) = {X, ф, {c}, {b, c}, {a, c}}; tri- 𝑏𝑔 ̂C(X) = {X, ф, {a}, {b}, {c}, {b, c}, {a, 
c}}; here {a} and {b} are tri- 𝑏𝑔 ̂closed sets but not a tri-g closed sets. 
 

Example 3.31 Let X = {a, b, c}, τ1 = {X, ф, {a}}, τ2 = {X, ф, {b}}, τ3 = {X, ф, {a, c}},       

tri- 𝑏𝑔 ̂C(X) ={X, ф, {b}, {c}, {b, c}, {a, c}}; tri- gC(X) ={X, ф, {b}, {c}, {a, b}, {b, 
c}, {a, c}}; here {a, b} is tri- g closed set but not a tri- 𝑏𝑔 ̂closed set. 

Remark 3.32 Tri- gs closed sets and tri- 𝑏𝑔 ̂closed sets are independent. 
 

Example 3.33 Let X = {a, b, c}, τ1 = {X, ф}, τ2 = {X, ф, {a, b}}, τ3 = {X, ф, {b, c}},           

tri- gsC(X) = {X, ф, {a}, {c}, {a, c}}; tri- 𝑏𝑔 ̂C(X) = {X, ф, {a}, {b}, {c}, {a, c}}; here 
{b} is tri- 𝑏𝑔 ̂closed set but not a tri- gs closed set. 
 

Example 3.34 Let X ={a, b, c}, τ1 = {X, ф}, τ2 = {X, ф, {a}}, τ3 = {X, ф, {b}}, tri- 

𝑏𝑔 ̂C(X) = {X, ф, {a}, {b}, {c}, {b, c}, {a, c}}; tri- gsC(X) = P(X);  here {a, b} is tri- 
gs closed set but not a tri- 𝑏𝑔 ̂closed set. 

Remark 3.35 The following diagram shows the relationship of tri- 𝑏𝑔 ̂closed sets with 
other known existing closed sets in tri- topological space.  

 

  

 

 

 

 

 

 

 

 

 

 

C 

B 

D 

H 

I 

E 

F 

G 

A 

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L. Jeyasudha, K. Bala Deepa Arasi 

 

A → Tri- 𝑏𝑔 ̂closed set B → Tri- closed set  C → Tri- b closed set 
D → Tri- g closed set  E → Tri- α closed set  F → Tri- gs closed set 

G → Tri- g*bw closed set H → Tri- bτ closed set I → Tri- semi closed set 

 

Remark 3.36 If (X, Tri- C(X)) is indiscrete topology then (X, Tri- 𝑏𝑔 ̂C(X)) is discrete 
topology but converse part need not be true. 

 

Example 3.37 Let X = {a, b, c}, τ1 = {X, ф, {a}}, τ2 = τ3 = {X, ф, {b, c}; Tri- C(X) = 

{X, ф, {a}, {b, c}}; Tri- 𝑏𝑔 ̂C(X) = {X, ф, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}} = P(X). 
Here, (X, Tri- 𝑏𝑔 ̂C(X)) is discrete topology but (X, Tri- C(X)) is not an indiscrete 
topology. 

Remark 3.38 If (X, Tri- C(X)) is discrete topology then (X, Tri- 𝑏𝑔 ̂C(X)) is discrete 
topology but converse part need not be true.  

 

Example 3.39 In example – 3.7, (X, Tri- 𝑏𝑔 ̂C(X)) is discrete topology but (X, Tri- 
C(X)) is not a discrete topology. 

Remark 3.40 If (X, Tri- C(X)) is indiscrete topology then, 

1) Every tri- 𝑏𝑔 ̂closed set is tri- b closed set. 
2) Every tri- 𝑏𝑔 ̂closed set is tri- g closed set. 
3) Every tri- 𝑏𝑔 ̂closed set is tri- gs closed set. 
4) Every tri- 𝑏𝑔 ̂closed set is tri- g*bω closed set. 
5) Every tri- g closed set is tri- 𝑏𝑔 ̂closed set. 
6) Every tri- gs closed set is tri- 𝑏𝑔 ̂ closed set. 
7) Every tri- bτ closed set is tri- 𝑏𝑔 ̂closed set. 
Example 3.41 Let X be any non-empty set, τ1 = τ2 = τ3 = {X, ф} are topologies of X.  

Tri- C(X) ={X, ф}; Tri- bC (X) = Tri- gC(X) = Tri- gsC(X) = Tri- bτC(X) = Tri- 

g*bωC(X) = Tri- 𝑏𝑔 ̂C(X) = P(X). 

Remark 3.42 If (X, Tri- C(X)) is discrete topology then, 

1) Every tri- 𝑏𝑔 ̂closed set is tri- closed set. 
2) Every tri- 𝑏𝑔 ̂closed set is tri- semi closed set. 
3) Every tri- 𝑏𝑔 ̂closed set is tri- α closed set. 
4) Every tri- 𝑏𝑔 ̂closed set is tri- b closed set. 
5) Every tri- 𝑏𝑔 ̂closed set is tri- g closed set. 
6) Every tri- 𝑏𝑔 ̂closed set is tri- gs closed set. 
7) Every tri- 𝑏𝑔 ̂closed set is tri- g*bω closed set. 
8) Every tri- g closed set is tri- 𝑏𝑔 ̂closed set. 
9) Every tri- gs closed set is tri- 𝑏𝑔 ̂closed set. 
10) Every tri- bτ closed set is tri- 𝑏𝑔 ̂closed set. 
 

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Tri – gb ˆ  Closed Sets in Tri- Topological Spaces 

 

 

 

Example 3.43 Let X be any non-empty set, τ1 = τ2 = τ3 = P(X) are topologies of X.    

Tri- C(X) = Tri- sC(X) = Tri- αC(X) = Tri- bC(X) = Tri- gC(X) = Tri- gsC(X) =        

Tri- bτC(X) = Tri- g*bωC(X) = Tri- 𝑏𝑔 ̂C(X) = P(X). 
 

 

4. Tri- 𝒃𝒈 ̂Open Sets In Tri- Topological Space  

Definition 4.1 The complement of a tri- 𝑏𝑔 ̂closed set is called the tri- 𝑏𝑔 ̂open set. The 
family of all tri- 𝑏𝑔 ̂open sets of X is denoted by tri- 𝑏𝑔 ̂O(X). 
 

Example 4.2 In example 3.2, tri- 𝑏𝑔 ̂O(X) = {X, ф, {a, b}, {b, c}, {a, c}}. 

Remark 4.3 ф and X are always tri- 𝑏𝑔 ̂open set. 

Remark 4.4 Intersection of tri- 𝑏𝑔 ̂open sets need not be tri- 𝑏𝑔 ̂open set. 
 

Example 4.5 In example – 3.2, tri- 𝑏𝑔 ̂O(X) = {X, ф, {a, b}, {b, c}, {a, c}}. Here, {a, 

b}, {b, c} are tri- 𝑏𝑔 ̂open sets but {a, b} ∩ {b, c} = {b} ∉ tri- 𝑏𝑔 ̂O(X). 

Remark 4.6 Union of tri- 𝑏𝑔 ̂open sets need not be tri- 𝑏𝑔 ̂open set. 
 

Example 4.7 In example – 3.16, tri- 𝑏𝑔 ̂O(X) = {X, ф, {a}, {b}, {c}, {a, b}, {a, c}}. 

Here, {b} and {c} are tri- 𝑏𝑔 ̂open sets but {b} ∪ {c} = {b, c} ∉ tri- 𝑏𝑔 ̂O(X). 

Remark 4.8 Difference of two tri- 𝑏𝑔 ̂open sets need not be tri- 𝑏𝑔 ̂open set. 
 

Example 4.9 In previous example – 4.7, tri- 𝑏𝑔 ̂O(X) = {X, ф, {a}, {b}, {c}, {a, b}, {a, 
c}}. Let A = X and B = {a}, Also A and B are tri- 𝑏𝑔 ̂open sets. But A\B = X\{a} = {b, 
c} is not a tri- 𝑏𝑔 ̂open set. 

Definition 4.10 Let (X, τ1, τ2, τ3) be a tri- topological space. The union of all tri- 

𝑏𝑔 ̂open sets of X contained in A is called the tri- 𝑏𝑔 ̂interior of A and is denoted by tri- 

𝑏𝑔 ̂int(A). (i.e) tri- 𝑏𝑔 ̂(A) = ∪ {B ⊆ X / B ⊆ A and A is tri- 𝑏𝑔 ̂open set}. 

Remark 4.11 

1)    tri- 𝑏𝑔 ̂int(ф) = ф,        
2)    tri- 𝑏𝑔 ̂int(X) = X,    

3)    tri- 𝑏𝑔 ̂int(A) ⊆ A,        

4)    tri- 𝑏𝑔 ̂int(A) = tri- 𝑏𝑔 ̂int(tri- 𝑏𝑔 ̂int(A)).  

Proposition 4.12 For any A ⊆X, (tri- 𝑏𝑔 ̂int(A))c =  tri-  𝑏𝑔 ̂cl(Ac). 

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Proof: (tri- 𝑏𝑔 ̂int(A))c = [∪ {G / G ⊆ A & G is tri- 𝑏𝑔 ̂open set}]c = ∩ { Gc / Gc ⊇ Ac 

& Gc is tri- 𝑏𝑔 ̂closed set} = ∩{ F / F ⊇ Ac& F is tri- 𝑏𝑔 ̂closed set} where F = Gc. 

Hence, (tri- 𝑏𝑔 ̂ int(A))c = tri- 𝑏𝑔 ̂cl(Ac). 

Proposition 4.13 Let (X,τ1,τ2,τ3) be a tri-topological space. Let A ⊆ X. Then tri- 

𝑏𝑔 ̂int(A) = A if  A is tri- 𝑏𝑔 ̂open set. 
Proof: Suppose A is a tri- 𝑏𝑔 ̂open set in X, then Ac is tri- 𝑏𝑔 ̂closed set in X. (i.e) tri- 

𝑏𝑔 ̂cl (Ac) ⊆ Ac. By the definition, Ac ⊆ tri- 𝑏𝑔 ̂cl(Ac). Therefore tri- 𝑏𝑔 ̂cl(Ac) = Ac ⇒ 

(tri- 𝑏𝑔 ̂int(A))c = Ac ⇒ tri- 𝑏𝑔 ̂int(A) = A.  

Remark 4.14 The tri- 𝑏𝑔 ̂interior of a set A is not always tri- 𝑏𝑔 ̂open set. 
 

Example 4.15 Let X = {a, b, c}, τ1 = {X, ф}, τ2 = τ3 = {X, ф, {a}}, tri- 𝑏𝑔 ̂C(X) = {X, 
ф, {b}, {c}, {a, b}, {b, c}, {a, c}}; tri- 𝑏𝑔 ̂O(X) = {X, ф, {a}, {b}, {c}, {a, b}, {a, c}}. 
Here, tri- 𝑏𝑔 ̂int ({b, c}) = {b, c} is not a tri- 𝑏𝑔 ̂open set. 

Proposition 4.16   

1) Every tri- open set is tri- 𝑏𝑔 ̂open set. 
2) Every tri- b open set is tri- 𝑏𝑔 ̂open set. 
3) Every tri- semi open set is tri- 𝑏𝑔 ̂open set. 
4) Every tri- α open set is tri- 𝑏𝑔 ̂open set. 
5) Every tri- g*bω open set is tri- 𝑏𝑔 ̂open set. 
6) Every tri- 𝑏𝑔 ̂open set is tri- bτ open set. 
Proof: By proposition – 3.17, 3.19, 3.21, 3.23, 3.25, 3.27 we get the results. 

 

5. Conclusions 

In this paper, we dealt with tri- 𝑏𝑔 ̂closed sets and tri- 𝑏𝑔 ̂open sets. In future we 
wish to do our research work in tri- 𝑏𝑔 ̂continuous functions, tri- 𝑏𝑔 ̂separated, tri-
 𝑏𝑔 ̂connected sets, tri- 𝑏𝑔 ̂ compact and so on. 

 

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