Generalized semi-extremally disconnectedness in double fuzzy topological spaces Academic research paper on "Mathematics"

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Title: Generalized semi-extremally disconnectedness in double fuzzy topological spaces

Author: Fatimah M. Mohammed M.S.M. Noorani A. Ghareeb

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S1658-3655(15)00032-1 http://dx.doLorg/doi:10.1016/j.jtusci.2015.0L008 JTUSCI153

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3-1-2015 24-1-2015

Please cite this article as: Fatimah M. Mohammed, M.S.M. Noorani, A. Ghareeb, Generalized semi-extremally disconnectedness in double fuzzy topological spaces, Journal of Taibah University for Science (2015), http://dx.doi.org/10.1016/j.jtusci.2015.01.008

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generalized semi-extremally disconnectedness in double fuzzy topological spaces

Fatimah M. Mohammed 1,2 * M. S. M. Noorani 1 and A. Ghareeb 3 1 School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia. 2 Permanent Address: College of Education, Tikrit University, Iraq. 3 Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt.

Abstract

In this paper we introduce the concepts of (r, s)-generalized fuzzy semi-extremally disconnectedness spaces and study effect of generalized double fuzzy semi-irresolute and generalized double fuzzy semiopen functions in this space. Moreover, we investigate some interesting relationship between generalized double fuzzy semiopen functions and (r, s)-generalized fuzzy semi-extremally disconnectedness spaces.

Keywords: double fuzzy topology, generalized double fuzzy semiopen function, (r, s)-generalized

fuzzy semi-extremally disconnectedness space.

AMS Subject Classification: 54A40, 45D05, 03E72.

1 Introduction

In 1968, the concept of fuzzy topological spaces was introduced by Chang[1]. In Chang's fuzzy topological spaces, each fuzzy set is either open or not. Later, the notion of an intuitionistic fuzzy sets which is a generalization of fuzzy sets were introduced by Atanassov [2], then Coker and coworker [3, 4] introduced the idea of the topology of intuitionistic fuzzy sets. Samanta and Mondal [5] gave the definition of an intuitionistic fuzzy topological space as a generalization of smooth topology of intuitionistic fuzzy sets. Working under the name "intuitionistic" did not continue and ended in 2005 by Garcia and Rodabaugh [6]. They concluded that work is under the name of "double". At a time not so long ago, [7] defined the ^p-operations on double fuzzy topological spaces.

Note that in recent years, topological space has been studied not only from the fuzzy point of view but also in the context of mixed topological spaces, bitopological spaces etc. (see [8], [9] and references there in.)

The class of L-fuzzy w-extremally disconnected spaces is defined by Sudha et al.[10]. Motivated by this, we now introduce the notions of generalized double fuzzy semi-extremally disconnectedness spaces and generalized double fuzzy semi-irresolute functions. Moreover, we investigate some interesting properties and characterizations of these concepts.

*E-mail addresses: nafea_y2011@yahoo.com(Fatimah M. Mohammed), msn@ukm.edu.my (M. S. M. Noorani), nasser-fuzt@hotmail.com(A. Ghareeb)

2 Preliminaries

Throughout this paper, X denotes a non-empty set, I the unit interval [0,1], I0 = (0,1] and I* = [0,1). The family of all fuzzy sets in X is denoted by IX. By 0 and 1, we denote the smallest and the greatest fuzzy sets on X. For a fuzzy set A G IX, 1 — A denotes its complement. Given a function f : X —> Y, f (A) and f-1(i) define the direct image and the inverse image of f, defined by f (A)(y) = Vf(x)=y A(x) and f-1(i)(x) = ¡(f (x)), for each A G IX, i G IY and x G X, respectively. All other notations are standard notations of fuzzy set theory.

Definition 2.1. [5, 6] A double fuzzy topology (t,t*) on X is a pair of maps t, t* : IX — I, which satisfies the following properties:

(01) t(A) < 1 — t*(A) for each A G IX.

(02) t(Ai A A2) > t(Ai) A t(A2) and t*(Ai A A2) < t*(Ai) V t*(A2) for each Ai, A2 G IX.

(03) t(Vier Ai) > Aier t(Ai) and t*(Ve Ai) < Vier t*(At) for each Ai G IX, i G r.

The triplet (X,t,t*) is called a double fuzzy topological spaces (dfts, for short). A fuzzy set A is called an (r,s)-fuzzy open ((r, s)-fo, for short) if t(A) > r and t*(A) < s, A is called an (r, s)-fuzzy closed ((r, s)-fc, for short) iff 1 — A is an (r, s)-fo set. Let (X,t1,t*) and (Y, t2,t2*) be two dfts's. A function f : X ^ Y is said to be a double fuzzy continuous iff T1(f-1(v)) > t2(v) and T**(f-1(v)) < t2*(v) for each v G IY.

Theorem 2.1. [11, 12] Let (X,t,t*) be a dfts. Then double fuzzy closure operator and double fuzzy interior operator of A G IX are defined by

CTiT*(A,r,s) = f\{i G IX | A < i, t(1 — i) > r, t*(1 — i) < s},

ITtT* (A, r, s) = y{i G IX I i < A, T(i) > r, T*(i) < s}, where r G I0 and s G I1 with r + s < 1.

Definition 2.2. [13] Let (X, t, t*) be a dfts. For each A, i G IX, r G I0 and s G I1.

(1) A fuzzy set A is called an (r, s)-generalized fuzzy semi-closed (briefly, (r, s)-gfsc) if CT,T* (A, r, s) < i, A < i, t(i) > r, t*(i) < s and i is (r, s)-fso set. A is called an (r, s)-generalized fuzzy semi- open (briefly, (r, s)-gfso) iff 1 — A is (r, s)-gfsc set.

(2) An (r, s)-generalized fuzzy semi-closure of A is defined by GSCT,T* (A, r, s) = A{i G IX | A < i and i is (r, s)-gfsc}.

(3) An (r, s)-generalized fuzzy semi-interior of A is defined by GSIT,T* (A, r, s) = V{i G IX | A < i and i is (r, s)-gfso}.

Definition 2.3. [14] Let (X, t1, t**) and (Y,t2, t2*) be dfts's. A function f : (X, t1,t**) (Y, t2,t**) is called:

(1) generalized double fuzzy semicontinuous (briefly, gdfsc) iff f-1(i) is (r, s)-gfsc set for each i G IY such that t2(1 — i) > r and t*(1 — i) < s.

(2) generalized double fuzzy semi-irresolute (briefly, gdfs-irr) if f-1(i) is (r, s)-gfso set for each (r, s)-gfso set i G IY, r G I0 and s G I*.

3 Properties and characterizations of generalized double fuzzy semi-extremally disconnected spaces

In this section, we introduced the concepts of generalized double fuzzy semi-extremally disconnected spaces. Some interesting properties and characterizations of the concepts are introduced and investigated.

Definition 3.1. Let (X,n,r1*) and (Y,T2,T2*) be dfts's. A function f : (X,n,r1*) ^ (Y,T2,t2*) is called generalized double fuzzy semi-open (briefly, gdfso) if f (A) is an (r, s)-gfso in Iy for each (r, s)-gfso set A E IX, r E I0 and s E I1,

Remark 3.1. Let (X, r, r*) be a dfts. For each A E Ix, r E I0 and s E I1, the following statements hold:

(1) GSIT,T* (A, r, s) = GSCr,t* (1 - A,r,s).

(2) GSCt,t* (A, r, s) = GSIt,t* (1 - A,r,s).

Proposition 3.1. Let (X, r1,r1*) and (Y, t2,t2*) be dfts's. A function f : (X, r1,r1*) ^ (Y, t2,t2*) is a gdfs-irr function iff f (GSCT1,t** (A, r, s)) < GSCT2,t** (f (A), r, s), for each fuzzy set A in IX, r E I0 and s E I1 .

Proof. Let A be any fuzzy set in Ix and f be a gdfs-irr function such that r E I0 and s E I1. Then, GSCT2,t** (f (A),r,s) is an (r, s)-gfsc set in IY. Since f is a gdfs-irr function so, f-1(GSCT2,t* (f (A),r,s)) is an (r, s)-gfsc set in Ix.

We have

A < f-1(f (A)) < f-1 (GSCt2,t**(f (A),r,s)). Also, by the definition of (r, s)-generalized fuzzy semi closure, we have

GSCti, t* (A, r, s) < f-1(GSCt2,t* (f (A),r,s)).

f (GSCti,t* (A, r, s)) < GSCt2 ,t* (f (A),r,s). Conversely, Let A be an (r, s)-gfsc set in IY such that

f(GSCtut1*(f-1(A),r,s)) < GSCti,t**(f(f-1(A),r,s)) GSCn,t1*(f-1(A),r,s) < f-1(A).

So that

f-1(A) = GSCti, t1* (f-1(A),r,s). That is, f-1(A, r, s) is an (r, s)-gfsc and hence, f is gdfs-irr funcion. □

Proposition 3.2. Let (X, r1,r1*) and (Y, t2,t2) be dfts's. If a function f : (X, r1,r1*) ^ (Y,t2 ,t2) is a gdfs-open surjective function, then for each fuzzy set A in Iy, r E I0 and s E I1,

f-1(GSCt2,t*(A,r,s)) < GSCT1,t**(f-1 (A),r,s).

Proof. Let A be any fuzzy set in Iy, r G I0 and s G I* such that i = f *(1 — A). Then

GSIT1, t* (f-1(1 — A),r,s) = GSIt! ,t* (i,r,s), is an (r, s)-gfso set in IX. But, GSIT1,t* (i,r,s) < i, hence

f (GSIti,t* (i,r,s)) < f (i).

gsit2 ,t** (f (gsit1, t* (i,r,s)),r,s) < gsit2 ,t* (f (i),r,s).

Since f is gdfs-open, so f (GSIT1 ,t* (i, r, s)) is an (r, s)-gfso in IY, r G I0 and s G I*. Therefore,

f (gsit1, t* (i,r,s)) < gsir* ,t** (f (i),r,s)

= GS *It2, t* (1 — A,r,s).

Hence,

GSIt1 t* (f-1(1 — A),r,s) = GSIt1,t** (i,r,s)

< f-1(GSIT2,T* (1 — A),r,s).

Therefore,

Hence,

Therefore,

1 — GSIt1 t* (f-1(1 — A),r,s) = 1 — GSItUt* (i,r,s)

> 1 — (f-*(GSIt2,t,* (1 — A,r,s))).

f-1(1 — (gsIt2,t**(1 — A,r,s))) < GSCtur*(1 — f-1(1 — A),r,s).

f-1(GSCr*T* (A, r, s)) < GSCT1 r* (f-1 (A),r,s).

Definition 3.2. A dfts (X,t,t*) is said to be an (r, s)-generalized fuzzy semi-extremely disconnected if GSCT,T* (A,r, s) is an (r, s)-gfso set for each A G IX, r G I0 and s G I* such that A is an (r, s)-gfso set.

Example 3.1. Let X = {a,b}. Define fuzzy sets A* and i* by:

A*(a) = 0.5, A*(b) = 0.5, i*(a) = 0.5, i-*(b) = 0.5.

Let (t,t*) defined by:

1, if A = 0 or1; t(A)={ *, if A = A*;

0, otherwise.

0, if A = 0 or! ; t*(A) = { *, if A = A*;

1, otherwise.

Since A* is an (*, *)-gfso set, also GSC(A*,r,s) = i* is an (*, *)-gfso set. Then (X,t,t*) is an (*, *)-generalized fuzzy semi-extremely disconnected space.

Proposition 3.3. Let (X,t1,t^) and (Y, t2,t2) be dfts's. If a function f : (X,t1,t^) ^ (Y,t2,r2*) is a gdfs-irr, gdfs-open surjective function such that (X, t\,t±) is an (r, s)-generalized fuzzy semi-extremely disconnected then, (Y, t2,t2*) is (r, s)-generalized fuzzy semi-extremely disconnected.

Proof. (1) Let A be an (r, s)-gfso fuzzy set in IY, r G I0 and s G I1 such that f is a gdfs-irr function, so f -1(A) is an (r, s)-gfso set in IX. But (X, t\,t± ) is an (r, s)-generalized fuzzy semi-extremely disconnected, then GSCTl,T*(f -1(A), r, s) is an (r, s)-gfso set in IX. Also, f is gdfs-open surjective function, GSCTl,T*(f-1(A),r, s)) is an (r, s)-gfso set in IY. Then, by Proposition 3.2,

and hence,

f-1(GSCt2,t*(A,r,s) < GSCti,<(f-1(a),r,s)

f-1(GSCt2,t* (A, r, s)) = GSCt2,t* (A, r, s)

< f(GSCtut*(f-1(A),r,s)

< GSCt2,t* (f(f-1(A),r,s)) = GSCt2,t* (A, r, s).

'2 T2*

which implies GSCT2,T* (A, r, s) = f (GSCTl,T* (f-1(A), r, s)). Therefore, GSCT2,T* (A, r, s) is an (r, s)-

gfso set in IY which implies (Y, t2,t2) is an (r, s)-generalized fuzzy semi-extremely disconnected.

' ' □

Theorem 3.1. Let (X,t,t*) be a dfts. Then the following conditions are equivalent:

(1) (X,t,t*) is an (r, s)-generalized fuzzy semi-extremely disconnected space.

(2) For each (r, s)-gfsc set A, GSIT,T* (A,r,s) is an (r, s)-gfsc set.

(3) For each (r, s)-gfso set A, GSCT'T* (A, r, s) V GSCT'T* (1 - GSCT'T* (A, r, s),r, s) = 1.

(4) For each (r, s)-gfso sets A and fi such that GSCT,T* (A, r,s) V fi = 1, then GSCT,T* (A, r, s) V GSCT, T * (f, r, s) = 1.

Proof. (1) ^ (2) Suppose A be a (r, s)-gfsc set. Then, 1 — A be a (r, s)-gfso set. So, GSCT,T* (1 — A,r,s) = 1 — GSITT* (A,r,s). By(1), GSCT'T* (1 — A,r,s) is an (r, s)-gfso, which implies that GSITT* (A,r,s) is an (r, s)-gfsc set.

(2) ^ (3) Suppose A be a (r, s)-gfso set. Then, 1 — A be a (r, s)-gfsc set. So by (2), GSIT,T* (1 — A,r,s) is a (r, s)-gfsc set. Hence

GSCt't* (A, r, s)VGSCt't* (1—GSCt't* (A, r, s),r, s) = GSCT'T* (A, r, s)VGSC''t* (GSIT'T* (1—A, r, s),r, s).

Therefore by Remark 3.1,

GSCt't * (A, r, s) V GSCt't * (1 — GSCt' t * (A, r, s), r, s) = GSCT' T * (A, r, s) V GSIt, t * (1 — A, r, s)

= GSCt' t * (A, r,s) V 1 — GSCt' t * (A, r, s)

(3) ^ (4) Suppose A and i be (r, s)-gfso sets such that

GSCr,t* (A, r, s) V i =1.

Then by (3),

1 = GSCr,r* (A,r,s) V GSCr,r* (1 — GSCr,t* (A,r, s),r, s).

i = 1 — GSCT,T* (A,r, s).

Therefore,

GSCr,r* (A, r, s) V GSCr,r* (i, r, s) = 1.

GSCr,t* (A, r, s) V GSCr,t* (i, r, s) = 1. which implies, GSCT,t* (A,r,s) is a (r, s)-gfso set so (X,t,t*) is an (r, s)-generalized fuzzy semi-

Theorem 3.2. A dfts (X,t,t*) is (r, s)-generalized fuzzy semi-extremely disconnected space iff for each a (r, s)-gfso set A and a (r, s)-gfsc set i such that A < i, GSCT,T* (A, r, s) < GSIT,T* (i, r, s).

Proof. Suppose (X,t,t*) is an (r, s)-generalized fuzzy semi-extremely disconnected space, A be a (r, s)-gfso set and i be (r, s)-gfsc set such that A < i. Then by Theorem 3.1, GSIT,T* (i,r,s) is an (r, s)-gfsc set. But by hypothesis, A is an (r, s)-gfso set such that A < i, this implies that A < GSIT,T* (i, r, s). Also, GSIT,T* (i, r, s) is an (r, s)-gfsc set which implies

Conversely, suppose i be any (r, s)-gfsc set. Then, GSIT,T* (i,r,s) is an (r, s)-gfso in IX and GSIT,T* (i,r,s) < i. Then,

is an (r, s)-gfsc set. By (2) of Theorem 3.1, (X, t, t*) is an (r, s)-generalized fuzzy semi-extremely

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