Induced fuzzy supra-topological spaces

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sets and systems Fuzzy Sets and Systems 91 (1997) 123-126

ELSEVIER

Induced fuzzy supra-topological spaces R.N. Bhaumik a'*, Anjan Mukherjee b aDepartment of Mathematics, Tripura University, Agartala 799 004, Tripura, India b R.K. Mahavidyalaya, Kailashahar 799277, Tripura, India Received May 1994; revised April 1996

Abstract The aim of this paper is to introduce and to study the concepts of induced fuzzy supra-topological spaces and s-lower semi-continuous functions, s-Lower semi-continuous functions turn out to be the natural tool for studying the induced fuzzy supra-topological spaces. © 1997 Elsevier Science Ltd. Keywords: Semi-open subset; Lower semi-continuity; Supra-topology; Fuzzy supra-continuity; Strong r-cut; Induced fuzzy topological space

1. Introduction

2. s-Lower semieontinuous functions

After the introduction of semi-open subsets by Levine [8], various concepts in topological space were introduced with the help of semi-open subsets instead of open subsets. In Section 2, the concept of s-lower semi-continuous function is introduced by using semi-open subsets. Some characterizations and properties of these functions are examined. In Section 3, these functions are used to define a new class of fuzzy supra-topological spaces, called induced fuzzy supra-topological spaces. The fuzzy supra-continuous functions and initial supratopological spaces are also investigated. The supra-interior and supra-closure of a fuzzy subset p are denoted respectively by/lsi and p~ [1].

Definition 1. A function f : (X, ~) - (R, tr) from a topological space (X, z) to a real number space (R, a) is said to be s-lower semi-continuous (resp. s-upper semi-continuous) at Xo ~ X iff for each e > 0, there exists a semi-open neighbourhood N(xo) such that x ~ N(xo) implies f(x) > f ( x o ) - e [resp. f ( x ) < f(xo) + 1.

* Corresponding author.

The following results can easily be proved analogous to the theorem in [2]. Result 1. The necessary and sufficient condition for a real-valued function f r o be s-lower semi-continuous is that for all r ~ R , the set {x ~ X : f ( x ) > r } is semiopen (or equivalently {x ~ X: f ( x ) r and 0 < r < 1} = a , ( f - 1(1A)).

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Thus f-I(1A) is fuzzy supra-open. Since f is fuzzy supra-continuous. By Theorem 2, a , ( f - l ( l a ) ) is semi-open in the topological space (X, z). T h u s f i s an irresolute function. Conversely, let f: (X, ~) --* (Y, rt) be an irresolute function and ~ is a fuzzy supra-open subset in (Y, Sa(rl)). Now for r > 0 , a , ( f - l ( a ) ) = { x e X : f - ~ (u(x)) > r} = (~ f ) ' (r, oo) = f - ~ (c~-a (r, oo)). Since ~ e Se(z~), c~ is s-lower semi-continuous and then ~-~(r, oo) is semi-open in (Y, ~). Also by hypothesis, f-~(~ ~(r, oo)) is semi-open in (X, ~), i.e. a,(f-a(cO) is semi-open in (X,v) which implies f-a(c~) eSe(v). Hence the theorem. [] Fuzzy supra-open function is defined in [1] as follows: A function f from a fuzzy supra-topological space (X,~) into a fuzzy supra-topological space (Y, ~ ) is called fuzzy supra-open i f f ( ~ ) ~ j for each cee ~ . We have the following theorem. Theorem 9. Let (X, ~(~1)) and (Y, ~(T2) ) be two induced fuzzy supra-topological spaces. If f:(X, oq'(rl )) ---,(Y, St(r2)) is an injectivefuzzy supra-continuous and fuzzy supra-open, then f : ( X , ~,(~,)) --+ (Y, ~ ) (~,)) is fuzzy continuous.

Proof. Let f:(X, Y(rl)) --+ (Y, 5~(Zz)) be an injective fuzzy supra-continuous supra-open function. If # e 5e(zl), then f(p)~Se(r2) by supra-open function. Now for each c¢~ r ( ~ , ) , ~c~f(l~)e,Y'(r2) by Theorem 6. Then f - l(~c~f(/~)) = f l (~)c~lt 5~(~1) by injective supra-continuity of f. Thus f - ~(~) e ~ ( ~ , ) for each ~ e Ye(~,) which proves that f: (X, o~(~,)) --, (Y, Y~f(~,)) is fuzzy continuous. [] 3.3 Initial supra-topology

Finally we shall define an initial supra-topology on X. Definition 4. Let (X, :T(z)) be an induced fuzzy supra-topological space. The family {a,(g): g 6 5e(v), r e l } of all semi-open subsets of X form a supratopology on X, called the initial supra-topology on X and is denoted by i(5~). (X, i(6e)) is called the initial supra-topological space. Thus the relation between the initial supra-topology and the corresponding topology r of 5a(z) is ~ c i(5e).

Example 2. Let X = {a, b, c, d} and ~ = {X, ~p, {c}, {d} {c, d}, {a, c, d} be a topology on X. Besides the members of z, {a, d}, {b, d} and {b, c, d} are also semi-open subsets in (X, ~). Since

{d} c {a,d} = {a,b,d} = Cl{d} {d} = {b,d} c {a,b,d} = Cl{d} d} = {b, c, d} = X = C l {c, d}.

Now 1~, Ix, l~c}, l{d}, llc.n ), l{a,c,a}, l{a,d}, lib, a} and llb.c.a> are s-lower semi-continuous, since the characteristic function of a semi-open subset is s-lower semi-continuous. Thus the collection of all these functions forms a induced fuzzy supra-topology on X. Here i(5a) = {cp, X, {c}, {d}, {c, d}, {a, d}, {b, d}, {b, c, d}, {a, c, d}}. Thus z = i(Sa). Note. If we take S0(X), the family of all semi-open subsets of X, then S0(X) = i(5").

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