INTERNATIONAL JOURNAL OF RESEARCH AND INNOVATION IN APPLIED SCIENCE (IJRIAS)

ISSN No. 2454-6194 | DOI: 10.51584/IJRIAS |Volume X Issue XI November 2025

b  I_S^Open Sets and Decomposition of Continuity Via Idealization

V. Jeyanthi

Dept. of Mathematics, Government Arts College for Women, Sivagangai - 630562, TamilNadu, India.

DOI: https://dx.doi.org/10.51584/IJRIAS.2025.101100094

Received: 04 December 2025; Accepted: 10 December 2025; Published: 19 December 2025

ABSTRACT

In this paper, we introduce the notion of b  I_S open sets and strong B_IS sets to obtain decomposition of

continuity via idealization. Additionally, we investigate properties of b  I_S open sets and strong B_IS sets

Key words and Phrases: semi -I_S- open sets, pre-I_S- open sets,



 I_S open sets, b  I_S open sets and strong

B_ISsets

AMS Subject Classification: 54A05, 54A10.

INTRODUCTION

Ideal in topological spaces have been considered since 1966 by Kuratowski[9]and Vaidyanathaswamy[16].After

several decades,in 1990, Jankovic and Hammlet [7] investigated the topological ideals which is the

generalization of general topology.Whereas in 2010,Khan and Noiri [8] introduced and studied the concept of

semi local functions.In 2014,Shanthi and Rameshkumar [14] introduced semi -I_S- open sets, pre-I_S- open sets

and



 I_S open sets.In this paper we introduce the notions of b  I_S open sets and strong B_IS sets to

obtain decomposition of continuity. Let X,





be a topological space and I is an ideal of subset of X. An ideal I

A I

B  A

implies

on a topological space X,



is a collection of nonempty subsets of X which satisfies (i)

and

B  I

A I

B  I

A  B  I

implies

and(ii)

and

.Given a topological space X, with an ideal I on X and if







:

















is the set of all subsets of X, a set operator

,called the local function of A with respect

and I , is defined as follows:for A  X, A^I,





{x X /U  A IforeveryU 







}

where

called











U 



/ xU



(Kuratowski 1966). A Kuratowski closure operator cl^





for a topology

 ^

I,





 

the

topology, finer than is defined by cl^





 A A^I,



(Vaidyanathaswamy,1945).When there is no



chance for confusion ,we will simply write A^for A^I,



and



^or

 ^



for

 ^

I,



.If I is an ideal on X,then

X,



, Iis called an ideal space.







G  A/G



, AIis a basis for



^(Jankovic and Hamlett,1992). If

A  X, cl(A) and int(A)will respectively denote the closure and the interior of A in X,





and int ^





will

denote the interior of A in



 ^



Definition1.1. LetX, be a topological space .A subset A of X is said to be semiopen[10] if there exists an

open set U in X such that U  A  clU.The complement of a semi open set is said to be semi-closed.The

collection of semi open(resp.semiclosed)sets in X is denoted by SO(X)(resp.SC(X)).The semi closure of A in (

X ,



)is denoted by the intersection of of all semiclosed sets containing A and is denoted by scl(A).

A  X

Definition1. 2. For

_I,

,where



x X /U  AIforeveryU SO(X)is called the semi-local function

[8] of A with respect to I and

SO(X, x) 



U SO(X) : xU



A_

instead of

A_



I,





We simply write

 ^s







U  E :U SO(X)andE I

.It is given in [1] that

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INTERNATIONAL JOURNAL OF RESEARCH AND INNOVATION IN APPLIED SCIENCE (IJRIAS)

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 ^s









U  X : cl^sX U  X U



.The closure operator cl^s



 A  A_and

denote

int^s





equivalently

the interior of the set A



 ^s

, I



It is known that





 ^







 ^s





Lemma1.3[8]. Let X,, I be an ideal topological space and

A, B  X

Then for the semi-local function the following properties hold:

A  B,then

(i)If

A  B_



(ii) If

,thenU  A_



U  A_

U 

Definition1.4.

A subset A of a topological space X is said to be

(i)

open [12] if A  int( cl(int( A)))





(ii)

pre-open [11] if A  int( cl(A))

(iii)

(iv)

(v)

semi-open [10] if A  cl(int( A)).

t-set [13] if int(A)=int(cl(A)).

b-open set [3] if A  int cl

clint



A .



(vi)

strong B-set [4] if

,where U is open, V is t-set and int(cl(A)) =cl(int(A)).

A U V

Definition1.5.

A subset A of an ideal topological space X,



, I is said to be

(i)

open [6] if A  int



cl^int









 I 

(ii)

(iii)

pre

open [5] if A  int



cl^





 I 

semi

open [6] if A  cl^int





 I 

(iv)

(v)

b  I  open [ 2] if A  int



cl^





cl^int



t-I-set [6]if int



cl^





 int

A .



(vi)

(vii)

B_I set [6] if

and V is a t-I-set.

A U V U 



Strong B_Iset [2] if

and V is a t-I-set andint



cl^





 cl^(int





A U V U 



Definition1.6.

X,, I

A subset A of an ideal space

is said to be

(i)

  I_S open [14] if A  int



cl^sint







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INTERNATIONAL JOURNAL OF RESEARCH AND INNOVATION IN APPLIED SCIENCE (IJRIAS)

ISSN No. 2454-6194 | DOI: 10.51584/IJRIAS |Volume X Issue XI November 2025

s

(ii)

pre  I_Sopen [14] if



A  int cl

 



(iii)

semi  I_Sopen [14] if A  cl^sint





(iv)

(v)

t  I_Sset [14] int

B_ISset [14] if



cl^s





 int





,where

and V is an t  I_Sset.

A U V

U 



The family of all



 I_S open (resp. Semi  I_S open, Pre  I_S open) sets an ideal topological space

X,



, Iis denoted by



ISO(X )(resp.SISO(X ), PISO(X )).

Lemma1.8[15].Let X,



, I be an ideal topological space and A  X

X,



, I

U cl^s





 cl^s



U  A

If U is open in

,then

2.b  I_Sopen set

Definition2.1A subset A of an ideal space X,



, I is said to be a b  I_Sopen set

if A  int



cl^s





cl^sint



A .



Proposition 2.1Le A be a b  I_Sopen set such that int( A) 



,then A is pre  I_Sopen.

Proof :Let A be a b  I_Sopen set. Then we have A  int



cl^s





cl^sint



A .



If int( A) 



,then cl^sint









.Therefore, A  int



cl^s





cl^sint



becomes

A  int



cl^s





Proposition 2.2 For a sub set of an ideal space X,, I

the following hold.

b  I_S

(i)Every open set is

open.

(ii)Every semi  I_Sopen set is b  I_Sopen.

(iii)Every pre  I_Sopen set is b  I_Sopen.

(iv)Every b  I_Sopen set is b-open.

Proof: (i),(ii),(iii) Obvious.

(iv)Let A be b  I_Sopen. Then we have

s

A  int



cl^s





 cl^int



 int



A^s A





int



  int



 int scl

 A sclint



 int





 int cl

clint



int



 int cl

clint





This shows that A is b-open.

Remark2.1. Converse of the Proposition 2.2 need not be true as seen from the following examples.

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Example 2.1Let





   



 and

    .Then

X  a,b,c,d







, a , b , a,b , a,d , a,b,d , X

I 



, b , c , b,c

(i)



is b  I_Sopen but it is not semi  I_Sopen.

A  a,b,c

(ii) A 



a,b,cis b  I_Sopen but it is not open.

Example2.2.Let X 



a,b,c















a,c, Xand I 





 

(i)

A 



b,cis b  I_Sopen but it is not pre I_Sopen.

Example 2.3. Let X 



a,b,c,d















a,d



a,b,d, Xand I 









Then A 



a,c,dis not b open but it is b  I_Sopen.

3.Strong B_ISset

Definition3.1. A subset A of an ideal space X,



, I is called strong B_IS set if A  U V, where

and

U 

V is a t  I_Sset and int



cl^s





 cl^sint





Proposition 3.1 Let X,



, I be an ideal space and A  X If A is a strong B_IS set, then A is a B_IS set.

Proof: Obvious

Remark 3.1. Converse of the Proposition 3.1 need not be true as seen from the following example.

Example 3.1Let X 



a,b,c















a,c, Xand I 







A 



,then int



cl^s

and cl^sint











and int













.Hence A is a t  I_S set.Clearly A is a B_IS set. But

int



cl^s



















.Hence int



cl^s





 cl^sint





So A is not a strong B_IS set.

Theorem3.1 Let X,



, Ibe an ideal space and A  X

.Then the following conditions are equivalent:

(i)

A is open;

A is b  I_Sopen and a strong B_IS set.

(ii)

Proof: (i)  (ii) By Proposition 2.2,every open set is b  I_Sopen.On the other hand every open set is strong

B_IS set, because X is t  I_Sset and int



cl^s





 cl^sint





(ii)  (i) Let A be b  I_S open and a strong B_IS set. Then

A  int



cl^s





cl^sint





 int



cl^s



U V



cl^sint



U V  where U is open and V is a t  I_S set and int



 cl^sint

Hence A  (int



cl^s





int



cl^s





) 



cl^sint



cl^sint







 A U 



int



cl^s





cl^sint







 A U int



cl^s





 A U int



 int



int



 int



U V int



A .



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So A is open.

Remark3.2. The notion of A is b  I_Sopenness is different from that of strong B_IS sets.

(i)

In Example2.1



cl^s





 cl^sint



  int







 .So A is a

A  b,c

 is not b  I_S open .But int

strong B_IS set.

(ii)

In Example2.1 A 



a,b,c



is b  I_S open .But int



cl^s





 X,cl^sint









a,b,d,int









a,b

So A is not a strong B_IS set.

Decomposition of continuity

Definition 4.1.A function f :X,







(Y,





is said to be b continuous[3] if for every V 



, f



^1V is b open

set of







B_IS

 I 

Definition 4.2. A function f :X,



,I



(Y,





is said to be

continuous[14](resp. semi

continuous

 I 

B 

 I 

[6], pre

continuous[5])if for every V 



, f



^1V is a

set(resp. semi

open set, pre

open

set)of





, I



  I_S

Definition 4.3. A function f :X,



,I



(Y,





is said to be

continuous[14](resp. semi  I_S

continuous [14], pre  I_S continuous[14])if for every V 



, f



^1V is an

open set(resp. semi  I_S



 I_S

open set, pre I_Sopen set)of X,, I

Definition 4.4. A function f :X,



,I



(Y,





is said to be b  I_S continuous,(resp.

strong B_IS

continuous) if for every V 



, f



^1V is a b  I_S set(resp. a strong B_IS set)of X,



, I

Proposition 4.1 Let X,



, I be an ideal space. If a function f :X,



, I









semi  I_S continuous(res. pre  I_S continuous),then f is b  I_S continuous.

Proof: This is an immediate consequence of Proposition 2.2 (ii) and (iii).

Proposition 4.2 Let X,



, I be an ideal space. If a function f :X,



,I



(Y,





b  I_S continuous ,then f is b continuous.

Proof: This is an immediate consequence of Proposition 2.2 (iv).

Proposition 4.3 Let



X,, I



be an ideal space. If a function f :











(Y,





strong B_IS continuous, then f is B_IS continuous.

Proof: This is an immediate consequence of Proposition 3.1 (i).

Theorem 4.1. Let X,, I be an ideal space. For a function f :X,,I



(Y,the following conditions are

equivalent:

(i)f is continuous;

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(ii) f is b  I_S continuous and strong B_IS continuous.

Proof: This is an immediate consequence of Theorem3.1.

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Math.J.,32,No 2(1992)273-284.

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Guler

and

G.Aslim,b-I-open

sets

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