SOME SEMI-REGULAR WEAKLY CONTINUOUS FUNCTIONS IN TOPOLOGICAL SPACES

* R. S. Wali, Basayya B. Mathad 1 and Nirani Laxmi 2 . 1. Department of Mathematics, Bhandari & Rathi College, Guledagudd-587 203, Karnataka, India. 2. Department of Mathematics, Rani Channamma University, Belagavi-591 156, Karnataka, India. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History


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vi) -open [11], if A is a finite union of regular open sets.The semi-pre-closure (resp.semi-closure, resp.pre-closure, resp.-closure) of a subset A of X is the intersection of all semi-pre-closed (resp.semi-closed, resp.pre-closed, resp.-closed) sets containing A and is denoted by spcl(A) (resp.scl(A),resp.pcl(A),resp.cl(A)).Definition 2.2 A subset A of a topological space X is called i) Generalized closed (briefly g-closed) [11], if ( ) whenever and U is open in X. ii) Generalized semi-closed (briefly gs-closed) [11], if ( ) whenever and U is open in X. iii) Generalized -closed (briefly g -closed) [11], if ( ) whenever and U is -open in X. iv) -generalized closed (briefly g-closed) [11], if ( ) whenever and U is open in X. v) Generalized semi pre-closed (briefly gsp-closed) [11], if ( ) whenever and U is open in X. vi) Regular generalized closed (briefly rg-closed) [11], if ( ) whenever and U is regular open in X. vii) Weakly closed (briefly w-closed) [11], if cl(A) U whenever A U and U is semi-open in X. viii) Regular weakly closed (briefly rw-closed) [11], if cl(A) U whenever A U and U is regular semi-open in X. ix) -regular weakly closed (briefly rw-closed) [12], if ( ) whenever and U is rw-open set in X.The complements of above all closed sets are their respective open sets in the same topological space X.

Definition 2.3
A subset A of a space X is said to be semi regular weakly closed (briefly srw-closed) set [10]
Lemma 2.7 Let ( ) be any topological space, in which i) Every semi-closed set is srw-closed set in X [10].
ii) Every rw-closed set is srw-closed set in X [10].
iii) Every srw-closed set is gs-closed set in X [10].iv) Every srw-closed set is gsp-closed set in X [10].)is called a rw-irresolute function [3], if the inverse image of every rwclosed set in ( ) for every rw-closed set V in ( ).

Lemma 2.11
A subset A of a topological space X is srw-open [11], if and only if ( ) whenever F is rwclosed and .

Semi-regular Weakly Continuous (briefly srw-continuous) Functions:-
In this section, we introduce the concept of srw-continuous functions in topological spaces and study their relations with various generalized continuous functions.We also discuss some properties of srw-continuous functions.and ( ) .Now ( ) , ( ) and (* +) * + are srw-closed sets in X.Hence, f is srw-continuous function.However, since {a, c, d} is not semi-closed set in X i.e. f is not semi-continuous on X.
) is completely continuous, then it is srw-continuous.
ii) Let F be closed subset of Y. Since f is -continuous, ( ) is a -closed in X.From theorem 3.8.3[10], ( ) srw-closed.Therefore f is srw-continuous.iii) Let F be closed subset of Y. Since f is -continuous, ( ) is a -closed in X.From Corollary 3.8.4 of [10], ( ) srw-closed.Therefore f is srw-continuous.iv) Let F be closed subset of Y. Since f is completely continuous, ( ) is a completely closed in X.From Corollary3.8.5 of [10], ( ) srw-closed.Therefore f is srw-continuous.and ( ) .Now ( ) , ( ) , (* +) * + and (* +) * + are srw-closed sets in X.Hence, f is srw-continuous function.However, since i) {b, c} is not -closed set in X i.e. f is not -continuous on X. ii) {b, c} is not -closed set in X i.e. f is not -continuous on X. iii) {b, c} is not -closed set in X i.e. f is not -continuous on X. iv) {b, c} is not regular closed set in X i.e. f is not completely continuous on X.  and ( ) .Now ( ) , ( ) and (* +) * + are gs-closed sets in X.Hence, f is gs-continuous function.However, since {c, d} is not srw-closed set in X i.e. f is not -continuous on X.

Corollary 3.12 If a function ( ) (
) is srw-continuous then it is gs-continuous.Proof: Let F be any closed subset of Y. Since f is srw-continuous, ( ) is a srw-closed in X.From theorem 3.10 of [10], ( ) gsp-closed.Therefore f is gsp-continuous.

Remark 3.13
The converse of Theorem 3.12 need not be true as shown from example 3.11 i.e. (* +) * + is gsp-closed set in X.Hence, f is gsp-continuous function.However, since {c, d} is not srw-closed set in X i.e. f is not -continuous on X.
Examples can be constructed to see that the concepts of g-continuity, w-continuity and -continuity are independent with the concept of srw-continuity.Thus the above discussion leads to the following implication diagram.

Remark 3 . 9
The converse of Corollary need not be true as shown in the following example.Let X= {a, b, c, d} with topology * * + * + * + + and Y= {p, q} with topology * * + * + +.Let function ( ) ( ) be defined by ( ) , ( ) , ( ) [10]rem 3.10 If a function ( ) () is srw-continuous, then it is gs-continuous, but not conversely.Proof: Let F be any closed subset of Y. Since f is srw-continuous, ( ) is a srw-closed in X.From theorem 3.4 of[10], every srw-closed set is gs-closed set but not conversely.i.e.( )gs-closed.Therefore f is gs-continuous.