Some more results on induced fuzzy topological spaces

FUZZY

sets and systems ELSEVIER

Fuzzy Sets and Systems 96 (1998) 255 258

Some more results on induced fuzzy topological spaces Anjan Mukherjee

Department of Mathematics, Ramkrishna Mahavidyalaya, Kailashahar, North Tripura 799277, India Received December 1995; revised September 1996

Abstract

The concept of induced fuzzy topological space, introduced by Weiss [J. Math. Anal. Appl. 50 (1975) 142 150], was defined with the notions of a lower semi-continuous function. The purpose of this paper is to present some new results on induced fuzzy topological spaces with the notion of ~-open sets introduced by Njastad [Pacific J. Math. 15 (1965) 961-970]. We also study the connections between some covering properties of an ordinary topological space (X, T) and its corresponding induced fuzzy topological space (X, W(T)). In the last section we also introduce a new class of fuzzy supra topological spaces - ~-induced fuzzy supra topological space (~-IFST space) and study the properties. @ 1998 Elsevier Science B.V. All rights reserved.

Keywords. Topology; Induced fuzzy topology; Fuzzy ~-compactness

1. Introduction

The concepts of fuzzy c~-open sets and fuzzy e-continuous functions were introduced by Mashhour, Ghanim and Fath Alla [4]. Further properties of these concepts were studied by Singal and Rajvanshi [11]. In Section 2, we study some more results on fuzzy e-sets. We denote the family of all c~-open subsets of topological space (X, T) by c~(X, T) and that of the fuzzy topological space (X, z) by F~(X, r). It is seen that r~ = {)L: ).C Int. Cl. Int. fi, fi c I x} is a fuzzy topology on X which has Fe(X, r) as subbase. The concept of induced fuzzy topological space (IFTS) was first introduced by Weiss [12]. Lowen [3] called these spaces as topologically generated spaces. The notion of lower semi-continuous (LSC) functions play an important role in defining the above concepts.

In Section 3, we prove that an IFTS (X, W(T)) is fuzzy e-compact iff (X, T) is e-compact. In the last section, we introduce a new class of fuzzy supra topological spaces, namely, e-induced fuzzy supra topological spaces (e-IFST space). These were defined with the generalized concept of e-continuous functions introduced by Noiri [9]. We also study the relation between the IFTS (X, W(T)) and that of e-IFST space (X, W~(T)). In this paper IA denotes the characteristic function of an ordinary subset A. We would like to mention the following definitions and results: (a) A fuzzy subset )~ of an fts (X, r) is said to be fuzzy e-open [4] if 2 C_Int. C1. Int. 2 or equivalently there exists a fuzzy open subset q such that q C )4 C Int. C1. ~/. (b) A function f : (X,r)---+(Y, r l ) from an fts (X, z) to another fts (Y, rl ) is said to be fuzzy e-continuous [4] (resp. fuzzy a-irresolute [10]) if f - I ( f l ) is fuzzy e-open in X for every ).~Zl (resp. ,;. is fuzzy e-open in Y).

0165-0114/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved PH S0165-0114(96)00308-9

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A. Mukherjee/Fuzzy Sets and Systems 96 (1998) 255-258

(c) Let (X,T) be a topological space. The collection W(T) of all lsc functions f : ( X , T ) ~ I forms a fuzzy topology on X. Then (X, W(T)) is known as an induced fuzzy topological space (IFTS)

Theorem 2.3. A fuzzy topological space (X,z) is fuzzy a-compact if and only if (X,z~) is fuzzy compact.

[5].

Proof. Let (X, z~) be fuzzy compact and let ~ = {2i, i E A} be an a-open cover of X. Then ~ is a r~-open cover of X, and hence ~ has a finite subcover. This shows that (X, r) is fuzzy a-compact.

(d) IrA E T then 1A E W(T) [12]. (e) 2 E W(T) iff for each r E I the strong r-cut cr~(2) E T [12] where at(2) = {x: 2(x) > r}.

The converse is a consequence of Alexander's subbase theorem for the topological space.

2. Some results on fuzzy ~-sets In this section we prove some more results on fuzzy-e open sets. Definition2.1. Let (X,z) be an fts, then z~= {2:2 C_Int. C1. Int. 2, 2 C I x } is always a fuzzy topology on X and satisfies z C z~, which has Fe(X, z) as subbase. Theorem 2.2. I f a function f : (X, z) ~ (Y,K) from

antis (X, r) to another fts (Y,K) is (a) fuzzy a-continuous then f : (X,z~) ~ (Y,K) is Juzzy continuous and (b) fuzzy e-irresolute then f : (X, r~) ~ (Y,K~) is fuzzy continuous. Proof. (a) Let 2 E K and f : (X, z) ~ (Y, K) be fuzzy e-continuous, then f - I ( 2 ) E F e ( X , z ) and thus f - l ( 2 ) E z ~ . Hence f " (X,~)--~ (Y,K) is fuzzy continuous. (b) Let /3 E K~ and f : (X,z) ~ (Y,K) be fuzzy n ~-irresolute, then /3 = Um(ni=l ~3ira) where ~3ira E Fe(Y,K), now, f - l ( / 3 ) = f - l ( U m Ni ,/3im) Um(ni~=lf-l(/3im)). Since f is fuzzy-irresolute f-l(/3im) E Fe(X,'c). This implies that f-1(/3) E z~ and hence f : (X, z~) ~ (Y, Ks) is fuzzy continuous.

Corollary 2.4. A subset 2 of a fuzzy topological space (X, r) is fuzzy ~-compact iff 2 &fuzzy compact in (X, r~). Theorem 2.5. Let f : (X,z) ~ (Y,K) be a fuzzy e-

continuous (resp. fuzzy a-irresolute) function. I f a subset 2 of X is r~-compact, relative to X then f ( 2 ) is fuzzy compact (resp. Ks-compact) relative to Y. Proof. Let {Gi: iE A} be a cover of f ( 2 ) by a k-open (resp. ks-open) subset in Y, then f - 1 ( 2 ) = {f-l(Gi): i E A} is a cover of 2 by z~-open subsets in X (By Theorem 2.2). Hence, by Corollary 2.4, )~ is r~-compact, so there exists a finite subset A o C A such that 2 c U { f - I ( G i ) : i E A0}. Thus f()~) C U{Gi: i E A0}. Hence f ( 2 ) is fuzzy compact (resp. K~-eompact) relative to Y.

3. Induced fuzzy topological spaces

=

Analogous to the Cheng's definition of fuzzy compactness, we define fuzzy e-compactness in the following way:

The concept of induced fuzzy topological space (IFTS) was first introduced by Weiss [12]. Lowen [3] called these spaces as topologically generated spaces and denoted them by W(T). In this section we study some more results on induced fuzzy topological spaces (IFT spaces). Result 3.1. I r A is :~-open in (X, T) then IA iS fuzzy

Definition 2.2. A fuzzy topological space (X,z) is fuzzy a-compact if every e-open covering of X has a finite subcover. Now we have the following theorem:

~-open in (X, W(T)). Proof. Let A be an a-open subset of (X, T). Then there exists an open subset B in (X, T) such that B C A C Int. C1. B. It follows that I8 C IA C I~nt.cL8.

A. Mukherjee/Fuzzy Sets and Systems 96 (1998) 255~58 • " IB C IA C Int. C1. IB [2, p. 1 13].

Now, IB ~ W(T), thus IA is fuzzy :z-open in (X, W(T)). Definition3.2. An IFTS ( X , W ( T ) ) is fuzzy :zcompact iff for each fuzzy :z-open family .~ C_ W ( T ) and for each a E (0, 1 ) such that Supac. e 6 ~>a, there exists a finite subfamily ~0 C ~ such that Sup6E:~ &~> a -- ~: where e ~ (0, a].

Using the above definition, we have the following theorem: Theorem 3.3. An IFTS (X, W ( T ) ) is Ji~zzy :z-compact iff the corresponding topological space (X, T) is :z-compact.

Proof. Let {A;: j C A} be an :z-open cover of X in T where A is the index set. Then the family of corresponding fuzzy :~-open subsets {1A/} of W(T) satisfies Supj 1Aj = 1. By fuzzy :z-compactness of (X, W ( T ) ) there exists J1,J2 . . . . . J~ such that S u p i l & ~ > l - c for 0 < e~~a > 0. We take b and 0 < ~:~< a such that a - e < b < a. Then {6-1(b, 1)}, ~ ,~ is a collection of open subsets and hence :z-open subsets which cover X; otherwise there exists an x E X such that for all & C ~', 6(X) <<,b < a, contradicting the fact that Sup6~e&~>a. Therefore there exists a finite sub collection {6i, i = 1,2, . . . , n } such that {&71(b,l]} covers X. By [2, L e m m a 3.1] {6yl[b, 1]} covers X. Thus Sup(6i)~> a - ~: which shows that (X. W ( T ) ) is fuzzy :z-compact.

257

:z-lower semi-continuous (:~-Lsc) [resp. :z-upper semicontinuous (:Z-USC)] at a point x0 of X iff for each > 0, there exists an :~-open neighbourhood N(xo) such that x E N(xo) implies f ( x ) > f(xo) - r (resp.

f ( x ) < f(xo) + ~). Now we have the following result: Result 4.2. (i) The necessary and sufficient condi-

tion for a real valued function f to be ~-Lsc is that Jor all rER, the set { x C Y : f ( x ) > r} is' ~-open. (ii) The function f from a space (X, T) to a space ( R, a r) where a ~ = (r, oc): r C R is :z-ls'c !ff the inverse image (~f ever)' open subset of (R, a ~) is :z-open in (x, r). (iii) The characteristic Junction of:z-open subsets" is" :z-lse. (iv) IJ" f~ is" a family of :z-Lsc Junctions then the function O, defined g(x) = sup~ fa(X) is :Z-Lsc. (v) I f f l , f 2 .... ,f~ are :z-Lsc functions, then the function h, defined by h(x)= infi:l Ji(x) is not :z-Lsc. Proof. The following results can easily be proved analogous to the theorems in [ 1]. Since every open set is a-open thus every Lsc function is a-Lsc but the converse is not true. Example 4.3. Let X = {a, b, c, d} and Y - {0, 1}, 7" = {qS, {c}, {d}, {a,c}, {c,d}, {a,c, d}, X } and 7"1 : {•, {1}, Y} be two topologies on X and Y, respectively. We define a function f : (X, T) ~ (Y, TI ) by f ( a ) : O , f ( b ) = f ( c ) = f ( d ) = l. H e r e f I ( 0 ) = {a}, f - ' ( 1 ) = {b,c,d}, f - ' ( Y ) = X we also ob serve that {e,d} C_{b,c,d} C_ Int. C l . { c , d } = X . Thus {b,c,d} is ~-open in (X, T). We fix r = 51 and using the result 4.2( l ) •/" i ( 1 ) : {b,c,d} = {x: f ( x ) > ½} is :z-open in (X, T) but not open in (X, T). Thus f is ~-lsc but f is not lsc.

4. ~-Induced fuzzy supratopological spaces

The aim of this section is to introduce a new class of fuzzy topological space - :z-induced fuzzy supra topological space (:Z-IFST space). These are defined with the notion of :z-lower semi-continuous (:z-Lsc) functions. Definition 4.1. A function f : X -+ R from a topological space to the real number space is said to be

Theorem 4.4. Let (X, T) be a topological space. The Jamily of all :~-Lsc Junctions Jrom the space (X, T) to the closed unit interval I = [0, 1] Jorms a .fuzzy supra topology on X.

Proof, Let W~(T) be the collection of all 2-Lsc functions from (X, T) to the unit closed interval I. We will now prove that W~(T) forms a fuzzy supra topology on X.

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(a) Since X is open, it is a-open and thus lx is ~-Lsc i.e. lx C W~(T). (b) q5 is c~-open thus 1~ is ~-Lsc i.e. 1~ E W~(T). (c) Let (2j) be an arbitrary family of c~-Lsc functions. Thus Sup{2j} is also ~-Lsc. Hence V 2j E W~(T). Thus W~(T) satisfies conditions (i)-(iii) of fuzzy supra topology. Definition 4.5. The fuzzy supra topology obtained as above is called c~-induced fuzzy supra topology (~-IFST) and the space (X, W~(T)) is called the a-induced fuzzy supra topological space (a-IFST space). The members of W~(T) are called fuzzy supra a-open subsets. Theorem 4.6. A Juzzy subset 2 in an ~-IFST space

(X, W~(T)) isJuzzy supra a-open ifffor each r c L the strong r-cut at(2) (resp. weak r-cut W~(2)) is a-open (resp. ~-closed) in the topological space (X, T). Proof. A fuzzy subset )~ is fuzzy supra c~-open in (X, W~(T)) i f 2 E W~(T) iff£ is ~-Lsc ifffor each r E I , {X E X: 2(x) > r) is ~-open in (X, T) i.e. 0-,.(2) is a-open in (X, T). Result 4.7. By Result 3.1 and Theorem 4.6, if A is ~-open in (X, T) then 1A 6 W~(T).

If W(T) is an I F T space and W~(T) is an ~-IFST space on X then W ( T ) C W~(T). Theorem 4.8.

Proof. Let 2 E W ( T ) i,e. 2 is a lower semi-continuous function. Since every Lsc function is ~-Lsc, 2 is c~-Lsc i.e. 2 E W~(T). Hence the theorem. Definition 4.9. A function f : (X,z) ---+ (Y, rl) from an ~-induced fuzzy topological space (X, z) to another ~-induced fuzzy topological space (Y, zl ) is said to be fuzzy supra ~-continuous if the inverse image of every fuzzy supra ~-open subset of Y is also fuzzy supra c~-open in X. Theorem 4.10. A function

f : (X, W~(T) ) ~ ( Y, W~(G)) is fuzzy supra ~-continuous iff f : (X,T) --+ (Y, G) is s-irresolute. Proof. Let f be fuzzy supra s-continuous and A is s-open in (Y, G) then

f-l(A):

{xEX: 1A(f(x)) = 1} = {xEX: f - l ( l A ( x ) ) > r, 0 < r<. 1} = ar(f-l(1A)).

Since A is c~-open in (Y, G), 1a E W~(G) (by Result 4.7). Thus, f - J ( 1A) is fuzzy supra ~-open in (X, W~(T)) (since f is fuzzy supra s-continuous), by Theorem 4.6. a r ( f - I ( 1 A ) ) is c~-open in (X,T). Thus f is ~irresolute. On the other hand, let us consider that f : (X, T) (Y,G) is s-irresolute and fl is a fuzzy supra ~open subset in (Y, W~(G)). Then for any P > 0, a p ( f - J ( f l ) ) = { x 6 X : f l(fl(x)) > p} = { x E X : I t ( f (x)) > p} = ( f l f ) - I ( p , oe) = f - l ( f l - l ( p , oc)). Now /~E W~(G). Thus // is 7-Lsc. Therefore 13-1(p, oo) is ~-open. Also by hypothesis f - l ( l ~ - l ( p , o~))is ~-open in (X, T)i.e. a p ( f - l ( f l ) ) is c~-open in (X, T) which implies f - l ( f l ) E W~(T). Hence f is fuzzy supra a-continuous.

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