# Generalized semi-closed functions and semi-generalized closed functions in bitopological spacesAcademic research paper on "Mathematics"

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{" ij-Semi-open set" / " ij-Semi-generalized closed set" / " ij-Generalized semi-closed set" / " ij-Semi-closure" / " ij-Semi T 1 / 2 -Space" / " ij-Semi-generalized functions" / " ij-Generalized semi-functions"}

## Abstract of research paper on Mathematics, author of scientific article — F.H. Khedr, H.S. Al-Saadi

Abstract In this paper we introduce and investigate the notions of a new class of generalized semi-closed functions and a class of semi-generalized closed functions in bitopological spaces. We study the further properties of ij-generalized semi closed and ij-semi-generalized closed sets. Applying of these concepts of sets, we introduce and study two new spaces, namely pairwise generalized s-regular and pairwise s-normal spaces.

## Academic research paper on topic "Generalized semi-closed functions and semi-generalized closed functions in bitopological spaces"

﻿Journal of the Egyptian Mathematical Society (2012) 20, 14-19

Egyptian Mathematical Society Journal of the Egyptian Mathematical Society

www.etms-eg.org www.elsevier.com/locate/joems

ORIGINAL ARTICLE

Generalized semi-closed functions and semi-generalized closed functions in bitopological spaces

F.H. Khedr a *, H.S. Al-Saadi b

a Department of Mathematics, Faculty of Sciences, University of Assiut, Assiut 715161, Egypt b Department of Mathematics, Faculty of Education for Girls, P.O. Box 4281, Makkah, Saudi Arabia

Available online 2 March 2012

KEYWORDS

//-Semi-open set; //-Semi-generalized closed set;

//-Generalized semi-closed set;

//-Semi-closure; //-Semi T1//2-Space; //-Semi-generalized functions; //-Generalized semi-functions

Abstract In this paper we introduce and investigate the notions of a new class of generalized semi-closed functions and a class of semi-generalized closed functions in bitopological spaces. We study the further properties of //-generalized semi closed and //-semi-generalized closed sets. Applying of these concepts of sets, we introduce and study two new spaces, namely pairwise generalized s-reg-ular and pairwise s-normal spaces.

© 2011 Egyptian Mathematical Society. Production and hosting by Elsevier B.V.

1. Introduction

In 1986 Fukutake [5] have generalized the notion of generalized closed set to bitopological spaces and defined a set A of a space (X, s1, s2) to be //-generalized closed (briefly, //-g-closed) set if/ — cl(A) C U, whenever A c U and U is s-open in X. Also, he defined a new closure operator and strongly pairwise T1/2-spaces.

* Corresponding author.

Peer review under responsibility of Egyptian Mathematical Society. doi:10.1016/j.joems.2011.12.008

In 1987, Bhattacharyya and Lahiri [2] defined the notion of semi-generalized closed sets by used the concept of semi-open set which defined by Levine [10].

In 1990 Arya and Nour [1] introduced the concept of generalized semi-closed sets using the semi-closure and studied some of their properties and characterizations of s-normal spaces. In 1993, Devi et al. [4] introduced sg-closed and gs-closed functions and study some of their properties.

The aim of this paper is to continue the study of generalized closed functions in bitopological spaces. We shall introduce //-sg-closed and //-gs-closed functions and we use the concepts of //-sg-closed and //-gs-closed sets to study the notions pair-wise generalized s-regular and pairwise generalized s-normal spaces. Also, we study further properties of //-generalized semi-closed, //-semi-generalized closed, //-gs-open and //-sg-open functions.

Throughout this paper (X, s1, s2), (Y, r1, r2) and (Z, v1, v2) (or briefly X, Y and Z) denote bitopological spaces on which no separation axioms are assumed unless otherwise mentioned. For a subset A of X, we shall denote the closure of A and the interior of A with respect to (or r,) by /-cl(A) and /-int(A) respectively for / = 1, 2. Also /, / = 1, 2 and /—/.

A subset A of a space X is said to be z/-semi-open [3], if there exists a s-open set U of X such that U c A c /— cl(U), or equivalently if A c j — cl(i — int(A)). The complement of an z/'-semi-open set is said to be z/'-semi-closed. The family of all z/'-semi-open sets of X is denoted by i/-SO(X) and for x 2 X, the family of all z/'-semi-open sets containing x is denoted by z/'-SO(X, x). An i/-semi-interior of A [3], denoted by i/-sint(A), is the union of all z/'-semi-open sets contained in A. The intersection of all z/'-semi-closed sets containing A is called the z/'-semi-closure of A [3] and is denoted by i/-scl(A). A subset A of X is said to be i/ — a-open [7] if A c i — int(/ — cl (i — int(A))).

Now, we mention the following definitions and results:

Definition 1.1. A subset A of a space X is called:

(i) An z/'-semi-generalized closed [9] (briefly z/'-sg-closed) if ji — scl(A) c U, whenever A c U and U 2 ij — SO(X). The complement of an i/-sg-closed set is called i/-sg-open. i/-sgint(A) and i/-sgcl(A) can be defined in a way similar to that of i/-sint(A) and i/-scl(A).

(ii) An i/-generalized semi-closed set [9] (briefly i/-gs-closed) if ji — scl(A) c U, whenever A c U and U is "-open in X. The complement of an i/-gs-closed set is called i/-gs-open. i/-gsint(A) and i/-gscl(A) can be defined in a way similar to that of i/-sint(A) and i/-scl(A).

Definition 1.2. A bitopological space (X, "1, s2) is said to be:

(i) Pairwise semi T 1=2-space [9] if and only if every z/'-sg-closed set is ji-semi-closed.

(ii) Pairwise normal [6] (resp. pairwise s-normal) if for any two disjoint "-closed set A and Tj-closed set B, there exist a Tj-open set U and a "-open set V (resp. U 2 ji — SO(X) and V 2 ij — SO(X)) such that A c U, B c V and U n V = / for i, j = 1,2, i-j.

(iii) Pairwise regular [6] if for each "-closed set F and a point x R F, there exist a "--open set U and a "-open set V such that x 2 U, F c V and U n V = / or, equivalently, for each "-open set U and x 2 U, there exists "-open set

V such that x 2 V c j — cl(V) c U.

(iv) Pairwise s-regular [8] if for each "-closed set F and a point x R F, there exist U 2 zj' — SO(X) and

V 2 ji — SO(X) such that x 2 U,F c V and U n V = /.

Definition 1.3 [9]. A function f: (X, s1, "2)!(Y, r1, r2) is called z/'-pre-semi-open (resp. z/'-pre-semi-closed) if f(U) is z/-semi-open in Y for every z/'-semi-open set U in X (resp. if f (U) is z/'-semi-closed for every z/'-semi-closed set U in X).

Lemma 1.4 [9]. f a function f: (X, s1, t2)!(Y, r1, r2) is z/'-pre-semi-c/osed, then for each subset S c Y and each U 2 i/ — SO(X) containing f" 1(S), there exists V 2 z/ — SO(Y) such that S c V and T1 (V) c U.

Theorem 1.5 [8]. A space X is pairwise s-regu/ar zf and on/y zf for each i-open set G and each x 2 G, there exists an z/'-semi-open set U such that x 2 U and/i — sc/(U) c G.

Theorem 1.6 [9]. If a function f: (X, s1, s2) ! (Y, r1, r2) is i-continuous and/i-pre-semi-c/osed, then f(A) is z/'-gs-c/osed set in Y, for every z/'-gs-c/osed set A of X.

2. Properties of //-generalized semi-closed mapping

Definition 2.1. A functionf: (X, s1, s2) ! (Y, r 1, r2) is called z/'-generalized closed (briefly, z/'-g-closed) if for each "--closed A of X, f(A) is z/'-g-closed set of Y. If f is 12-g-closed and 21-g-closed, then f is called pairwise g-closed.

Definition 2.2. A function f: (X, s1, s2) ! (Y, r1, r2) is called z/'-generalized semi-closed (briefly, z/'-gs-closed) if for each "closed set A of X, f(A) is an z/'-gs-closed set in Y. If f is 12-gs-closed and 21-gs-closed, then f is called pairwise gs-closed.

Theorem 2.3. Every /-c/osed map is z/'-g-c/osed and every z/-g-c/osed map is z/'-gs-c/osed.

Proof. Follows directly, since every /-closed set is z/'-g-closed and every z/'-g-closed set is z/'-gs-closed. □

Theorem 2.4. Every /z-semi-c/osed map is z/'-gs-c/osed map.

Proof. Since every /i-semi-closed set is i/-gs-closed set, the theorem follows.

The converses of the above theorems are not true in general as shown in the following. □

Example 2.5. Let X ={a, b, c}= Y, s1 ={/, {a}, fb, cg, Xg, "2 = {/, {b}, {a, b}, X}, r1 = {/, {a}, Y}, r2 = {/, {a}, {a, b}, Y}. Let f : (X, s1, s2) ! (Y, r1, r2) be the identity function. Then f is 12-g-closed and 12-gs-closed but not 2-closed and hence not 21-semi-closed, since {a, c} is s2-closed set in X but f({a, c}) = {a, c} is not 21-semi-closed in Y.

Example 2.6. An i/-gs-closed map need not be i/-g-closed.

Let X = Y = {a, b, c}, "1 = {/, {a}, X}, "2 = {/, {b, c}, X}, r1 — {/, {a, b}, Y} and r2 = {/, {a}, Y}. If a map f: (X, t1, t2)!(Y, r1, r2) be defined by f(a)—b,f(b) —a, f(c) — c. Set A — {a} is s2-closed. Then A is 12-gs-closed in Y, but f(A) — {b} is not 12-g-closed, since 2 — cl({b}) — {b, c}{a, b} 2 r1.

Example 2.7. An z/'-gs-closed map need not be ji-semi-closed.

Let X — {a, b, c}, Y — {a, b, c, d}, s1 — {/, {b, c}, X}, t2 — {/, {a}, X}, r1 —{/, {c, d}, Y}, r2 — {/, {a, b}, Y} and f: (X, t1, t2) ! (Y, r1, r2) be a map defined by f(a) — a,f(b) — b,f(c) — d. Then f(X) — {a, b, d} is 12-gs-closed in Y but not 21-semi-closed in Y, since 21-scl({a, b, d}) — Y.

Theorem 2.8. A function f: (X, s1, t2)!(Y, r1, r2) is z/'-gs-c/osed if and on/y if for each subset S c Y and each "/-open set U containing f—1(S), there is an z/'-gs-open set V c Y such that S c Vand f—1(V) c U.

Proof. Let S c Y and U be a "-open set containing H1(S). Then V — Y \f(X \ U) is z/'-gs-open set of Y containing S and f—1 (V) c U. □

Conversely, suppose that for each S c Y and a "-open set U containing f—1(S), there is an z/'-gs-open set V such that S c V and f—1(V) c U. Let F be a "-closed set of X and W

be a rropen set of Y such that f(F) c W. Then r1 (Y\f(F)) c X\ Fand X\ Fis -¡/-open. By hypothesis, there is an //-gs-open set V such that Y \f(F) c V and r1 (V)cX\ F. Therefore, F c X\ f—1 (V) and hence f(F) C Y \ V. Since Y \ W c Y \ f(F), f—1(Y \ W) c f—1 (Y \ f(F)) c i—1(V)cX\ F. By taking complements, we get F c X \ f—1(V) cX \ f—1 (Y \ f(F)) c X \ f—1(Y \ W). Therefore, f(F) c Y \ V c W. Since Y \ V is //-gs-closed set and // — sclf(F)) c// — scl(Y\ V) c W implies // — sclf(F)) c W and hence f(F) is //-gs-closed. Thus f is //-gs-closed.

Theorem 2.9. Let a function f: (X, s1, s2) ! (Y, r1, r2) be i-continuous and //-gs-closed. If A is an //-g-closed set of X, then f(A) is //-gs-closed.

Proof. Let f(A) c U, where U is r-open set of Y. Then A c n1(U). Since f is /-continuous, then H1(U) is a ¡¡-open set in X. Hence / — cl(A) c f~ 1(U), since A is //-g-closed set. Therefore, f(/ — cl(A)) c U. Since f is //-gs-closed, then f(/-cl(A)) is //-gs-closed. Thus // — sclf(/ — cl(A))) c U. On the other hand f(A) cf(/ — cl(A)) implies // — scl(f(A)) c// — sclf(/ — cl(A))) c U. Therefore,f(A) is //-gs-closed. □

Remark 2.10. Let f: (X, s1, s2) ! (Y, r1, r2) and g : (Y, r1, r2) ! (Z, v1, v2) are two functions, then:

(i) If f is /-closed and g is //-gs-closed, then gof is ij-gs-closed.

(ii) If f is //-g-closed and g is /-continuous, //-gs-closed, then gof is //-gs-closed.

(iii) If f is //-gs-closed and g is /-continuous, //-pre-semi-closed, then g is //-gs-closed.

(iv) If f is a /-continuous surjection and gof is //-gs-closed, then g is //-gs-closed.

(i) Let U be a ¡ -closed set of X. Since f is /-closed, thenf(U) is a rj-closed set of Y. Since g is //-gs-closed, then (gof)(U) = g(f(U)) is an //-gs-closed set. Hence gof is an //-gs-closed function.

(ii) Let U be a /-closed set of X. Since f is //-g-closed, then f(U) is an //-g-closed set of Y. By Theorem 2.9, (gof)(U) = g(f(U)) is //-gs-closed set. Hence gof is //-gs-closed.

(iii) Let U be a ¡-closed set of X. Since f is //-gs-closed, then f(U) is a //-gs-closed set of Y. By Theorem 1.6, we have (gof)(U) = g(f(U)) is //-gs-closed. Hence gof is an //-gs-closed function.

(iv) Let V be a rj- closed set of Y. Since f—1 (V) is ¡-closed in X, then (gof)(f—1 (V)) is a //-gs-closed set of Z. Thus g( V) is //-gs-closed. Hence g is an //-gs-closed function. □

Theorem 2.11. Iff: (X, s1, s2)! (Y, r1, r2) is an y-gs-closed function, then for every subset A c X, // — gscl(f(A)) c f/ — cl(A)).

Proof. Let A be a subset of a space X. Since f is /j-gs-closed, f(/-cl(A)) is //-gs-closed set containing f(A). Hence // — gscl (f(A)) c / — gscl f(/ — cl(A))) = f(/ — cl(A)). □

Definition 2.12. A function f: (X, s1, s2) ! (Y, r1, r2) is called //-generalized semi-open (briefly //-gs-open) if for each ¡¡-open set U of X, f(U) is //-gs-open set in Y.

Theorem 2.13. For a function f: (X, s1, s2) ! (Y, r1, r2):

(/) f/s//-gs-open

(ii) f (J — int(A)) c ij — gsint(f (A)) for each subset A of X.

(iii) For each x 2 X and for each /-open set U conta/n/ng x, there /s an //-gs-open set V conta/n/ng f(x) such that

V c f (U).

(/v) f f /s sur/ect/ve, thenf—1 (ij — gscl(B)) c J — cl(f~l (B)) for each subset B of Y. Then (/) ) (//) ) (m) ) (/v).

(i)) (ii): Let A be a subset of a space X. Since J — int(A) c A, then f (J — int(A)) c f (A). But /-int(A) is ¡-open set of X, then f(/-int(A)) is /j-gs-open set of Y, since f is /j-gs-open. Hence f (J — int(A)) = ij — gsintf (J — int(A))) c ij— gsint(f (A)). Thus f (J — int(A)) c ij— gsint (f (A)).

(ii) ) (iii): Let x 2 X and U be a ¡-open set containing x.

Then by (ii), f (J — int(U))c ij — gsint (f (U)) and this implies f (U) c ij— gsint(f (U)). Thus there exists an /j-gs-open set V such that f (x) 2 V and V c f (U).

(iii) ) (iv): Let B c Y and x 2 f—1(ij — gscl(B)). Then

f (x) 2 ij — gscl(B). If x R J — cl(f"1 (B)), then x 2 U where U = X \ J — cl(f—1(B)). Then by (iii), there is an /j-gs-open set V such that f (x) 2 V c f (U). Now, V c f (U) c f (X\ (r11 (B))) c Y \ B which shows that B c Y \ V. Since Y \ V is //-gs-closed, ij — gscl(B) c Y \ V. Now, f (x) 2 ij — gscl(B). Hence f (x) R V which is contradiction. Thus f—1(ij — gscl (A))c J — cl(f—1(B)). □

Theorem 2.14. If a funct/on f: (X, ¡1, ¡2) ! (Y, r1, r2) /s ij-gs-open, then for each subset B c Y and each ¡j-closed set F containing f 1(B) , there is an //-gs-closed set V c Y such that B c V and f—1(V) c F.

Proof. Let B c Y and F be a ¡--closed set containing f—1 (B). Then V — Y \ f(X \ F) is //-gs-closed set of Y containing B

and f—1(V) c F. □

3. Properties of //-generalized semi-closed functions

Definition 3.1. A functionf: (X, ¡1, ¡2) ! (Y, r1, r2) is called //-semi-generalized closed (briefly //-sg-closed) if for each ¡/-closed set F of X, f(F) is //-sg-closed set in Y. Iff is 12-sg-closed and 21-sg-closed, then f is called pairwise sg-closed.

Theorem 3.2. If f: (X,¡1,¡2)!(Y, r1, r2) is i/-sg-closed function, then i/ — sgclf(A)) c f(/ — cl(A)) for every subset A c X.

Proof. Let A be a subset of a space X. Since f is i/-sg-closed function, then we have f(/-cl(A)) is an i/-sg-closed set containing f(A). Hence z/ — sgclf(A)) c z/ — sgclf(/ — cl(A))) — f(/ — cl(A)). □

Definition 3.3. A functionf: (X, t1, "2) ! (Y, r 1, r2) is called z/-semi-generalized open (briefly i/-sg-open) if for each "/-open set U of X, f(U) is z/'-sg-open set in Y.

Theorem 3.4. For a function f: (X, "1, "2) the fo//owing:

(Y, r1, r2), consider

(i) f is ij-sg-open

(ii) f (j — int(A)) c ij — sgint(f (A)) for each subset A of X.

(iii) For each x 2 X and for each j-open set U containing x, there is an ij-sg-open set V containing f(x) such that V c f (U).

(iv) if f is surjective, then f—1 (ij — sgcl(B)) c j — c/(f-1 (B)) for each subset B of Y. Then (i) ) (ii) ) (iii) ) (iv).

(i) ) (ii): Let A be a subset of a space X, then /-int(A) is a "j-open set of X. Since j — int(A) c ij— sgint(A) c A, then f (j — int(A)) c f (j— sgint(A)) c f (A). By (i), f(/-int(A)) is z/'-sg-open set of Y. Hence f (j — int(A)) c ij — sgint (f (ij — sgint(A))) — f (ij — gsint(A)). Thus f (j — int(A)) c ij— sgint(f (A)). (ii) ) (iii): Let x 2 X and U be a "j-open set containing x.

Then by (ii), f (j — int(U))c ij — sgint(f (U)) and this implies f (U) c ij — sgint(f (U)). Thus there exists an i/-sg-open set V such that f (x) 2 V and V c f (U). (iii)) ) (iv): Let x R j — cl(f—1(B)), then x 2 U where U — X \ j — cl(f—1(B)). Then by (iii), there is an z/'-sg-open set V such that f (x) 2 V c f (U). Now, V c f (U) c f (X \(f—1 (B))) c Y \ B which shows that B c Y \ V. Since Y \ V is z/'-sg-closed, ij — sgcl(B) c Y \ V. Moreover, T1 (ij — sgcl(B)) c X \ f—1(V) and hence x R f—1(ij- — sgcl(B)). Thus T1 (ij — sgcl(B)) c j — cl(f—1 (B)). □

Theorem 3.5. A functionf: (X, "1, "2) ! (Y, r1, r2) is an z/'-sg-open, B c Y and F is ",-c/osed set containing f—1 (B), then there is an z/'-sg-c/osed set V c Y such that B c V and P1 (V) c F.

Proof. Similar to that of Theorem 2.14. □

Remark 3.6. Let f: (X, "1, "2)!(Y, r1, r2) and g: (Y, r1, r2) ! (Z, v1, v2) be two functions, if f is /-continuous surjection and gof is i/-sg-closed, then g is i/-sg-closed.

Proof. It similar to the proof of Remark 2.10 (iv). □

Since every ji-semi-closed set is i/-sg-closed, we can state the following theorem.

Theorem 3.7. Every /i-semi-c/osed function is i/-sg-c/osed.

Example 3.8. An i/-sg-closed functon need not be ji-semi-closed.

Let X, Y, "1 and "2 be as in Example 2.7, let r1 — {/, {a, d}, {a, b, d}, Y} and r2 — {/, {a, b, c}, Y}. Let f: (X, "1, "2) ! (Y, r1, r2) be the identity function. Set X is "2-closed, also f(X) — X. Then X is 12-sg-closed in Y but X is not 21-semi-closed in Y, since 21-scl(X) = Y.

Since every i/-sg-closed set is i/-gs-closed, we can state the following theorem.

Theorem 3.9. Every z/'-sg-c/osedfunction is z/'-gs-c/osed.

Example 3.10. An z/'-gs-closed function need not be z/'-sg-closed.

Let X, Y, "1, "2 and r1 be as in Example 2.7, let r2 — {/, {a, b}, Y} and f: (X, "1, "2)!(Y, r1, r2) be a map defined by f(a) — a,f(b) — c,f(c) — d. Set X — {a, b, c} is "2-closed andf(X) — {b, c, d} is 12-gs-closed butf(X) is not 12-sg-closed, since 21 — sclf(X)) — Y{a, c, d}2 12 — SO( Y).

j-closed

ji-semi-closed ij-g-closed

ij-sg-closed

ij-gs-closed

4. Pairwise generalized s-regular and pairwise generalized s-normal spaces

Definition 4.1. A bitopological space (X, "1, "2) is said to be a pairwise generalized s-regular (briefly pairwise gs-regular) space if for each i/-gs-closed set F and a point x R F, there exists disjoint i/-semi-open set U and ji-semi-open set V such that x 2 U, F c V and U n V — /.

Lemma 4.2.

Pairwise Pairwise Pairwise

regular ) s — regular ( gs — regular space space space

Proof. It follows from that every "i-closed set is ji-gs-closed and every i-open set is i/-semi-open.

One may give examples to show that implications in Lemma 4.2 may not be revisable. □

Theorem 4.3. f f: (X, "1, "2) ! (Y, r1, r2) is a pairwise continuous, z/'-semi-open and z/'-gs-c/osed sur/ection from a pairwise regu/ar space X to a space Y, then Y is pairwise s-regu/ar.

Proof. Let U be a ri-open set containing a point y in Y. Suppose that x be a point of X such that y = f(x). Since X is pair-wise regular space, then there is a "i-open set V such that x 2 Vc j — cl(V) c f—1(U), where f—1(U) is ",-open set since f is pairwise continuous. Then y 2f( V) c f(j — cl( V) c U

and /(j-cl(F)) is i/-gs-closed, since f is i/-gs-closed. Therefore, we have ji — scl/j — cl(V))) C U. Moreover, y 2f(V) C ji— sclf(V)) C U and f(V) is i/-semi-open in Y, since / is i/-semi-open. Hence by Theorem 1.5, Y is pairwise s-regular. □

Corollary 4.4. Let/: (X, s1, s2) ! (Y, r 1, r2) be a pairwise cow-tiWMOMs, i/-semi-opew awd i/-sg-c/osed sMr/ectiow. If X ij a pair-wise regular space, thew Y is pairwise s-regu/ar.

Proof. The proof is obvious from Theorem 4.3 and the fact that every i/-sg-closed map is i/-gs-closed. □

Theorem 4.5. A space X is pairwise gs-regu/ar if and ow/y if for each i/-gs-opew set U and each x 2 U, there exists aw i/-semi-opew set V such that x 2 V awd/i — sc/( V) C U.

Proof. Let Xbe a pairwise gs-regular space, U be an ij-gs-open set and x 2 U. Then X \ U is i/-gs-closed and x R X \ U. There exists an i/-semi-open set V and a ji-semi-open set H such that x 2 V, X\ U C H and Vn H = /. Then X \ H C U and

V C X \ H. Since X\ H is ji-semi-closed, ji — scl(V)CX \ H. Then x 2 V and ji — scl( V) C U. □

Conversely, let F be an i/-gs-closed set and x R F. Then X\ F is an ij-gs-open set and x 2 X\ F. There exists i/semi-open set V such that x 2 V and ji — scl(V)CX\ F. Since ji-scl(V) is ji-semi-closed, then H = X\ji — scl(V) is a ji-semi-open set such that F C H, also Vn H = /. Thus X is pairwise gs-regular.

Definition 4.6. A space X is said to be pairwise generalized s-normal (briefly pairwise gs-normal) if for each pair of i/-gs-closed set A and ji-gs-closed set B such that A n B — /, there exist disjoint U 2 ji — SO(X) and V 2 i/ — SO(X) such that A C U, B C V and U n V — /.

Lemma 4.7.

Pairwise Pairwise Pairwise

normal ) s — normal ( gs — normal space space space

Proof. It follows from that every s-closed set is ji-gs-closed and every i-open set is i/-semi-open. □

One may give examples to show that implications in Lemma 4.7 may not be revisable.

Theorem 4.8. A space X is pairwise gs-worma/ if awd ow/y if for every /i-gs-c/osedset Fawd i/-gs-opew set G cowtaiwiwg F, there is aw i/-semi-opew set V such that F C V C /i — sc/( V) C G.

Proof. Let F be a ji-gs-closed and G be an ij-gs-open set such that F C G. Then X \ G is i/-gs-closed. There exist U 2 ji — SO(X) and V 2 i/ — SO(X) such that X\ G C U and F C V such that U n V — /. Then X \ U C G and U C X \ V or V C X\ U. Since X\ U is ji-semi-closed, also ji — scl( V) C X \ U andji — scl(V)CG. □

Conversely, let A be i/-gs-closed and B be ji-g-closed sets of X. Then X \ A is ij-gs-open contained B. There exists

V 2 i/ — SO(X) such that B C Vandji — scl(V) C X\ A. Since

ji-scl(V) is ji-semi-closed, then U — X\ ji — scl( V) is a ji-semi-open set such that A c U, also U n V — /. Thus X is pairwise gs-normal space.

Definition 4.9. A function f: (X, "1, "2) ! (Y, r1, r2) is called z/'-generalized semi-irresolute (briefly ij-gs-irresolute) if f—1(V) is z/'-gs-closed set for each z/'-gs-closed set V of X. If f is 12-gs-irresolute and 21-gs-irresolute, then f is called pairwise gs-irresolute.

Theorem 4.10. Let X be a pairwise gs-norma/ space and /et f:(X, "1, "2) ! (Y, r1, r2) be a pairwise gs-irreso/ute, pairwise presemi-c/osedfunction from X onto a space Y. Then Y is pair-wise gs-norma/ space.

Proof. Let A be i/-gs-closed set and B be ji-gs-closed set of Y. Since f is pairwise gs-irresolute, f—1(A) is z/'-gs-closed and f—1(B) is ji-gs-closed. Since X is pairwise gs-normal, then there exist disjoint U 2 ji — SO(X) and V 2 z/ — SO(X) such that n1 (A) c U and F1 (B) c V. Now f is pairwise pre-semi-closed, therefore there exist G 2 ji — SO( Y) and H 2 z/ — SO( Y) such that A c G, B c H with f—1(G) c U and f—1(H) c V, by Lemma 1.4. Since U n V — /, then G n H — /. Hence Y is pairwise gs-normal. □

Theorem 4.11. Let X be a pairwise gs-regu/ar space and /et /• (X, "1, "2) ! (Y, r1, r2) be a pairwise gs-irreso/ute, pairwise pre-semi-c/osedfunction from X onto a space Y. Then Y is pairwise gs-regu/ar.

Proof. Similar to that of Theorem 4.10. □

Theorem 4.12. Iff: (X, "1, "2) ! (Y, r1, r2) is a pairwise continuous, pairwise gs-c/osed sur/ection from a pairwise norma/ space X to a space Y, then Y is pairwise s-norma/.

Proof. Let A be a ri-closed set and B be a r/-closed set of Y such that A n B — /. Since f is pairwise continuous, f—1(A) is ",-closed set and H1(B) is "/-closed set in X such that T1 (A)n f—1 (B) — f—1(A n B) — /. Since X is pairwise normal, there exist disjoint "/-open set U and "i-open set V of X such that T1 (A) c U and f—1(B) c V. By Theorem 2.8, there exist ij-gs-open set G and ji-gs-open set H of Y such that A c G, B c H, f—1(G) c U and f—1(H) c V. Then we have T1 (G)n f—1 (H) — / and hence G n H — /. Thus Y is pairwise s-normal. □

Theorem 4.13. Iff: (X, "1, "2) ! (Y, r1, r2) is a pairwise continuous, pairwise gs-c/osed sur/ection from a pairwise regu/ar space X to a space Y, then Y is pairwise s-regu/ar.

Proof. Similar to that of Theorem 4.12. □

Corollary 4.14. Let f : (X, "1, "2) ! (Y, r1, r2) be a function. Then:

(i) If f be a pairwise continuous, pairwise semi-c/osed sur/ection and X is pairwise norma/, then Y is pair-wise s-norma/.

(ii) If f be a pairwise continuous, pairwise sg-closed sur-jection and X is pairwise normal( resp. pairwise regular), then Y is pairwises-normal (resp. pairwise s-regular).

Proof. (i) and (ii): The proof is obvious from Theorem 4.11 and a fact that every ji-semi-closed function is j-gs-closed and every j-sg-closed function is j-gs-closed. □

Corollary 4.15. Let f : (X, s1, s2) ! ( Y, r1, r2) be a function. Then:

(i) If f be a pairwise continuous, pairwise semi-closed surjection and X is pairwise regular, then Y is pair-wise s-regular.

(ii) If f be a pairwise continuous, pairwise sg-closed sur-jection and X is pairwise regular, then Y is pairwise s-regular.

Proof. Similar to that of Corollary 4.14. □

References

[1] S.P. Arya, T.M. Nour, Characterization of s-normal spaces, Indian J. Pure Appl. Math. 8 (21) (1990) 717-719.

[2] P. Bhattacharyya, B.K. Lahiri, Semi-generalized closed sets in topology, Indian J. Math. 29 (3) (1987) 375-382.

[3] S. Bose, Semi open sets, semi continuity and semi open mapping in bitopological spaces, Bull. Calcutta Math. Soc. 73 (1981) 237246.

[4] R. Devi, H. Maki, K. Balachandran, Semi-generalized closed maps and generalized semi-closed maps, Mem. Fac. Sci. Kochi Univ. Math. 14 (1993) 41-54.

[5] T. Fukutake, On generalized closed sets in bitopological spaces, Bull. Fukuoka Univ. Ed. III 35 (1986) 19-28.

[6] J.C. Kelly, Bitopological spaces, Proc. Lond. Math. Soc. 3 (13) (1963) 71-89.

[7] F.H. Khedr, ca-Continuity in bitopological spaces, Arab. J. Sci. Eng. 17 (1) (1992) 85-89.

[8] F.H. Khedr, T. Noiri, s-Closed bitopological spaces, J. Egypt. Math. Soc. 15 (1) (2007) 79-87.

[9] F.H. Khedr, H.S. Al-Saadi, On pairwise semi-generalized closed sets, J. King Abdul Aziz Univ. Sci. 20 (4) (2008), September.

[10] N. Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo 19 (2) (1970) 89-96.