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Published in Fuzzy Sets and Systems, 1996, Vol 80, Iss 3, P 377- Archived with Dspace@nitr381 author email abehera@nitrkl.ac.in http://dspace.nitrkl.ac.in/dspace On fuzzy compact-open topology S. Dang a, A. Behera b* Department of Mathematics, GovernmentCollege, Rourkela-769 004, India bDepartment of Mathematics, Regional Engineering College, Rourkela-769 008, India Received October 1994 Abstract The concept of fuzzy compact-open topology is introduced and some characterizations of this topology are discussed. Keywords: Fuzzy locally compact Hausdorff space; Fuzzy product topology; Fuzzy compact-open topology; Fuzzy homeomorphism; Evaluation map; Exponential map. 1. Introduction 2. Preliminaries Ever since the introduction of fuzzy set by Zadeh The concepts of fuzzy topologies are standard by [13] and fuzzy topological space by Chang [2], now and can be referred from [2, 8, 13]. For the several authors have tried successfully to generalize definitions and various results of fuzzy topology, numerous pivot concepts of general topology to the fuzzy continuity, fuzzy open map, fuzzy compact- fuzzy setting. The concept of compact-open topol- ness we refer to [2]. For the definitions of fuzzy ogy has a vital role in defining function spaces point and fuzzy neighborhood of a fuzzy point we in general topology. We intend to introduce the refer to [8]. concept of fuzzy compact-open topology and con- So far as the notation is concerned, we denote tribute some theories and results relating to this fuzzy sets by letters such as A, B, C, U, V, W, etc. I x concept. The concepts of fuzzy local compactness denotes the set of all fuzzy sets on a nonempty set and fuzzy product topology play a vital role in the X. 0x and lx denote, respectively, the constant theory that we have developed in this paper. We fuzzy sets taking the values 0 and 1 on X. A, A t, A have used the fuzzy locally compactness notion due will denote the fuzzy closure, fuzzy interior and the to Wong [11], Christoph [3] and fuzzy product fuzzy complement of A e I x, respectively. topology due to Wong [12]. We need the following definitions and results for our subsequent use. Theorem 2.1 (Ganguly and Saha [5]). A mapping f : X ~ Y from antis X to antis Y is said to be f u z z y * Corresponding author. continuous at a f u z z y point xt of X if and only if for
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every fuzzy neiohborhood V of f(xt), there exists a fuzzy open set U such that xt e U and U is fuzzya fuzzy neighborhood U of x, such t h a t f ( U ) <~ V. compact. Since X is fuzzy Hausdorff, by Theorem f is said to be fuzzy continuous on X if it is so at 4.13 of [6], U is fuzzy closed; thus U = 0. Considereach fuzzy point of X. a fuzzy point Yr ~ (Ix - U). Since X is fuzzy Haus- dorff, by Definition 2.3 there exist open sets C andDefinition 2.2. (Chang [2]). A mapping f: X ~ Y D such that xt ~ C and y, ~ D and C A D = 0x. Letis said to be a fuzzy homeomorphism iffis bijective, V = C A U. Hence V~<U implies I T ~ < u = u .fuzzy continuous and fuzzy open. Since 17 is f-closed and U is fuzzy compact, by Theorem 4.2 of [6], it follows that 17 is fuzzy com-Definition 2.3 (Srivastava and Srivastava [10]). An pact. Thus x~ e 17 ~< U and 0 is fuzzy compact. Thefts (X, T) is called a fuzzy Hausdorff space or T2- converse follows from Proposition 2.6(b). []space if for any pair of distinct fuzzy points (i.e.,fuzzy points with distinct supports) xt and y,, there Definition 2.8 (Katsara and Lin [7]). The productexist fuzzy open sets U and V such that xt ~ U, of two fuzzy sets A and B in an fts (X, T) isy~Vand U A V=Ox. defined as (A Ã— B)(x, y ) = min(A(x), B(y)) for all (x, y) e X Ã— Y.Definition 2.4 (Wong [11]). An fts (X, T) is said tobe fuzzy locally compact if and only if for every Definition 2.9 (Azad [1]). I f f / : X i ~ Yi, i = 1,2,fuzzy point xt in X there exists a fuzzy open set then fl xf2:X1 Ã— X 2 ----r Y1 Ã— Y2 is defined byU e T such that xt ~ U and U is fuzzy compact, i.e., (fl Ã— f 2 ) ( x l , x 2 ) = (fl(xl),f2(x2)) for each (xl,x2)each fuzzy open cover of U has a finite subcover. X1 Ã— X2.Note 2.5. Each fuzzy compact space is fuzzy locally Theorem 2.10 (Azad [1]). For any two fuzzy setscompact. A and B in I x , ( A Ã— B ) = ( A x l x ) V ( l x x B ) . Similarly, if fi : Xi ~ Yi, i = 1, 2, thenProposition 2.6. In an Hausdorff fts X, the following (fl Ã—f2)-l(Bx Ã— B : ) = f ? l ( B 1 ) Ã— f 2 1 ( B 2 ) for allconditions are equivalent: BI Ã— B2 c X2 x Y2. (a) X is fuzzy locally compact. (b) For each fuzzy point xt in X , there exists Definition 2.11. Let X and Y be two ftss. The a fuzzy open set U in X such that xt ~ U and 0 is function T : X Ã— Y ~ Y x X defined by T (x, y) =fuzzy compact. (y, x) for each (x, y)~ X x Y is called a switching map.Proof. (a) =*- (b): By hypothesis for each fuzzypoint xt in X there exists a fuzzy open set U which The proof of the following lemma is easy.is fuzzy compact. Since X is fuzzy Hausdorff (afuzzy compact subspace of a fuzzy Hausdorff space Lemma 2.12. The switchino map T : X x Yis fuzzy closed [6]) U is fuzzy closed: thus U = 0. Y Ã— X defined as above is fuzzy continuous.Hence xt e U and 0 is fuzzy compact. (b) =, (a): Obvious. [] 3. Fuzzy compact-open topologyProposition 2.7. Let X be a Hausdorfffts. Then X isfuzzy locally compact at a fuzzy point xt in X if and We now introduce the concept of fuzzy compact-only if for every f-open set U containing xt, there open topology in the set of all fuzzy continuousexists an f-open set V such that xt ~ V, 17 is fuzzy functions from an fts X to an fts Y.compact and 17 <<.U. Definition 3.1. Let X and Y be two ftss and letProof. First suppose that X is fuzzy locally com-pact at an f-point xt. By Definition 2.4 there exists y x = {f: X ~ Y I f is fuzzy continuous}.
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We give this class yX a topology called the fuzzy We now consider the induced map of a givencompact-open topology as follows: Let function f : Z x X -~ Y.:,f = {K e IX: K is fuzzy compact in X} Definition 4.3. Let X, Y, Z be ftss and f : Z x X ~ Y be any function. Then the inducedand m a p f : X - * r z is defined by (f(xt))(zr)=f(zr, xt)~= VelY:VisfuzzyopeninY}. for f-points x~ ~ X and zr ~ Z. Conversely, given a function f : X - - , y z a corresponding functionFor any K ~ :U and V e ~, let f can also be defined by the same rule. The continuity of f can be characterized in termsNr.v = { f â€¢ Y X : f ( K ) <~ V}. of the continuity o f f and vice versa. We need the following result for this purpose.The collection {NK, v: K e ~ , V â€¢ U } can be usedas a fuzzy subbase [4] to generate a fuzzy topology Proposition 4.4. Let X and Y be two fiss withon the class yx, called the fuzzy compact-open Y fuzzy compact. Let xt be any fuzzy point in X andtopology. The class Y x with this topology is called N be a fuzzy open set in the fuzzy product spacea fuzzy compact-open topological space. Unless X x Y containing {xt} x Y. Then there exists someotherwise stated, yX will always have the fuzzy fuzzy neighborhood W of xt in X such thatcompact-open topology. {x,} x Y ~ W x Y <~N. Proof. It is clear that {xt} x Y is fuzzy homeomor-4. Evaluation map phic [2] to Y and hence {x,} x Y is fuzzy compact [9]. We cover {x~} x Y by the basis elements We now consider the fuzzy product topological {U x V} (for the fuzzy topology of X x Y) lying inspace y x x X and define a fuzzy continuous map N. Since {x,} x Y is fuzzy compact, {U x V} hasfrom y x x X into Y. a finite subcover, say, a finite number of basis elements UI x V1 . . . . . U, x V,. Without loss ofDefinition 4.1. The mapping e: y X x X --, Y de- generality we assume that x t ~ U i for eachfined by e(f, xt) = f ( x t ) for each f-point xt ~ X and i = 1,2 . . . . . n; since otherwise the basis elementsf s yX is called the fuzzy evaluation map. would be superfluous. LetTheorem 4.2. Let X be a fuzzy locally compact W=%Ui. i=1Hausdorff space. Then the fuzzy evaluation mape: yX x X -~ Y is fuzzy continuous. Clearly W is fuzzy open and xt e W. We show thatProof. Consider (f, x,) e yX x X where f â€¢ yX and W x Y ~ + (Ui x Vi). i=1x t E X . Let V be a fuzzy open set containingf(x~) = e(f, xt) in Y. Since X is fuzzy locally com- Let (xr, Ys) be any fuzzy point in W x Y. We con-pact and f is fuzzy continuous, by Proposition 2.7, sider the fuzzy point (x,,ys). Now (xt, y~)e Ui x Vithere exists a fuzzy open set U in X such that for some i; thus Yse Vi. But x r e Uj for everyx, â€¢ U, 0 is fuzzy compact a n d f ( U ) ~ V. j = 1, 2 . . . . . n (because x, e W). Therefore ( x , ys) e Consider the fuzzy open set Nc.v x U in yX x X. Uix Vi, as desired. But Uix Vi <~ N for all i = 1,Clearly ( f , x , ) â€¢ N v . v x U. Let ( g , x , ) e N c . v X U 2, ..., n; andbe arbitrary; thus 9(t2)~< V. Since x , e U, wehave g(x,)e V and hence e ( g , x , ) = g ( x , ) e V . W x Y <~ + (Uix Vi). i=lThus e ( N c . v x U ) < ~ V , showing e to be fuzzycontinuous. [] Therefore W x Y ~< N. []
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Theorem 4.5. Let Z be a fuzzy locally compact Theorem 5.1 (The exponential law). Let X and Hausdorff space and X, Y be arbitrary fuzzy Z be fuzzy locally compact Hausdorffspaces. Then topological spaces. Then a map f: Z Ã— X -4 Y is for any fuzzy topological space Y, the functionfuzzy continuous if and only if ~ X - 4 y z is fuzzy continuous, where f is defined by the rule E: y z , , x ._, ( y z ) x(f(x~))(z~) =f(z~, x,). defined by E ( f ) = f (i.e., E(f)(xt)(z,,) = f(z,,, xt)= (f(xt))(z,,))for all f: Z x X -4 Y is a fuzzy homeomor-Proof. Suppose that f is fuzzy continuous. Con- phism.sider the functions Proof. (a) Clearly E is onto.ZxX .... ~ Z x YZ ~ , y Z x Z , y, (b) For E to be i~ective, let E ( f ) = E(g) for f, g: Z Ã— X -4 Y. Thus f = 0, where f a n d 0 are the induced maps o f f and g, respectively. Now for any where iz denotes the fuzzy identity function on Z, fuzzy point x, in X and any fuzzy point zu in Z, t denotes the switching map and e denotes the f(z,,, xt) = (f(xt)(z,,)) = (0(xt)(z,)) = g(z,,, xt); thus evaluation map. Since et(iz xf)(z~, x,)= et(z~,f(x,)) = e(f(x,),z~)---J~xt)(z~)=f(z~, xt), it follows that f=g. (c) For proving the fuzzy continuity of E, con-f = et(iz x f); and f being the composition of fuzzy sider any fuzzy subbasis neighborhood V of f in continuous functions is itself fuzzy continuous. ( y z ) x i.e., V is of the form N~:.w where K is a fuzzy Conversely suppose that f is fuzzy continuous. compact subset of X and W is fuzzy open in yZ. Let Xk be any arbitrary fuzzy point in X. We have Without loss of generality we may assume thatf(xk) e y z . Consider a subbasis element Nr, v= W = NL. v where L is a fuzzy compact subset of {g e yZ: o(K) <~ U, K ~ I z is fuzzy compact and Z and U E I r is fuzzy open. Thenf(K) ~< NL.V = W U e I r is fuzzy open} containing f(Xk). We need to and this implies that (f(K))(L) <~ U. Thus for any find a fuzzy neighborhood [8] W of Xk such that fuzzy point x, in K and for all fuzzy point z, in L, wef ( w ) <~NK, v; this will suffice to prove f to be have (f(x,))(z,,) ~ U, i.e.,f(z,, x,) ~ O and therefore a fuzzy continuous map. f ( L Ã— K)<~ U. Now since L is fuzzy compact in For any fuzzy point z, in K, we have Z and K is fuzzy compact in X, L x K is also fuzzy (f(Xk))(Z,) =f(z,,, Xk) ~ U; thus f ( K x {Xk}) ~ U, compact in Z Ã— X (cf. [9]) and since U is a fuzzy i.e., K x {Xk} <~f-I(U). Sincefis fuzzy continuous, open set in Y we conclude t h a t f e NLÃ—K.C, ~< yzÃ—x.f - ~ ( U ) is a fuzzy open set in Z x X. Thus f - I(U) is We assert that E(NLÃ—K.V) <~ Nr.w. Let g ~ NLÃ—K,u a fuzzy open set in Z x X containing K x {Xk}. be arbitrary. Thus g(L Ã— K)<~ U, i.e., g(z,,, x t ) = Hence by Proposition 4.4, there exists a fuzzy (0(xt))(z,) ~ U for all fuzzy point z, e L ~< Z and for neighborhood W of xk in X such that K x {xk} all fuzzy points xt ~ K <<.X. So (0(xt))(L) ~< U for ~KxW ~f-I(U). Therefore f ( K x W ) ~ U . all fuzzy points xt e K ~< X, i.e., (0(xt)) E NL, V = W Now for any x, e W and z , ~ K , f ( z ~ , x , ) = for all fuzzy points x , ~ K <~X, i.e., (0(x~))~ (f(x,))(z,)e U. Therefore (f(x,))(K)<<, U for all NL, v = W for all fuzzy points xt ~ K <~X. Hence x , e W, i.e., f(x.)e NK.v for all x , e W. Hence we have 0(K)~< W, i.e., 0 = E ( g ) e Ni,:,w for anyf ( w ) ~ NK, v as desired. [] g e NLÃ—r,u. Thus E(NLÃ—K.v)<~ Nicw. This proves that E is fuzzy continuous. (d) For proving the fuzzy continuity of E- 1 we5. Exponential map consider the following evaluation maps: ej :(yZ)X x X - - * Y z defined by e l ( f x t ) = f ( x t ) where We define exponential law by using induced f e ( y Z ) X and x, is any fuzzy point in X andmaps (of. Definition 4.3) and study some of its e2 : yZ x Z -4 Y defined by e2(g, z,,) = g(z,) whereproperties. g e yZ and z, is a fuzzy point in Z. Let 0 denote the
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c o m p o s i t i o n of the following fuzzy c o n t i n u o u s Proof. C o n s i d e r the following c o m p o s i t i o n s :functions: X x yX x Z r r , yXxzrxx ,Ã—i, , z r x y x x X(ZxX)Ã—(yZ)X "r ( y Z ) X x ( Z x X ) :--. z r x ( y X x X ) iÃ—"1, Z r x Y ": , Z, iÃ—, ( y z ) x x ( X x Z) where T, t d e n o t e the switching m a p s , ix a n d i de- note the fuzzy identity functions on X a n d Z r, respectively, a n d ej a n d e2 d e n o t e the e v a l u a t i o n ((yZ)XxX)x z c.,.z yZxZ ~ y, maps. Let q~ = e2 ~(i x el) Â° (t x ix)" T. By T h e o r e m 5.1, we have an e x p o n e n t i a l m a pwhere i, iz d e n o t e the fuzzy identity m a p s on yXxZ r E:ZX~Ã—z " __. (Z x)(yZ)X a n d Z respectively a n d T , t d e n o t e theswitching maps. T h u s O : ( Z x X) x (yZ)X __. y , i.e., zXÃ— ~Ã—z ( z X ) r , Ã—z Since q~ , E(q~) e ; let S = E(q~),~b ~ Y IzÃ—x~Ã—ly~x. W e c o n s i d e r the m a p i.e., S : y X x z r ~ z x is fuzzy c o n t i n u o u s . F o r~ : yczÃ—xl~Iv~l x ~ ( y z Ã— x ff~l " f 6 y X , g ~ Z r a n d for any fuzzy p o i n t xt in X, it is easy to see that S ( f g)(xt) = g ( f ( x t ) ) . [ ](as defined in the s t a t e m e n t of the t h e o r e m , in fact itis E). So ReferencesE(qJ) ~ ( y z Ã—x )IY~p, [1] K.K. Azad, On fuzzy semi-continuity, fuzzy almost conti-i.e., we have a fuzzy c o n t i n u o u s m a p nuity and fuzzy weakly continuity, J. Math. Anal, Appl. 82 (1981) 14 32.p,(t~):(yz)x ~ yz,~x. [2] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182 190. N o w for a n y fuzzy p o i n t s z , , e Z , x t e X and [3] F.T. Christoph, Quotient fuzzy topology and local com- pactness, J. Math. Anal. Appl. 57 (1977) 497-504.f e y z Ã— x , it is a r o u t i n e m a t t e r to check t h a t [4] D.H. Foster, Fuzzy topological groups, J. Math. Anal. (E(tp)o E ) ( f ) ( z , , xt) = f ( z , , xt); hence /~(~b)oE= Appl. 67 (1979) 549-564. identity. Similarly for a n y O e ( Y Z ) x a n d fuzzy [5] S. Ganguly and S. Saha, A note on 6-continuity and 6- p o i n t s x, e X, z, E Z, it is also a r o u t i n e m a t t e r to connected sets in fuzzy set theory, Simon Stevin 62 (1988) check t h a t (EÂ°F,(O))(O)(x,)(zu)= O(xt)(z,); hence 127- 141. [6] S, Ganguly and S. Saha, A note on compactness in fuzzy E o/~(~) = identity. This c o m p l e t e s the p r o o f that setting, Fuzzy Sets and Systems 34 (1990) 117 124. E is a fuzzy h o m e o m o r p h i s m . [] [7] A.K. Katsara and D.B. Liu, Fuzzy vector spaces and fuzzy topological vector spaces, J. Math. Anal. Appl. 58 (1977)Definition 5.2. T h e m a p E in T h e o r e m 5.1 is called 135-146. [8] Pu Pao Ming and Liu Ying-Ming, Fuzzy topology I: neigh-the e x p o n e n t i a l m a p . borhood structure of a fuzzy point and Moore Smith con- vergence, J. Math. Anal. Appl. 76 (1980) 571 599. A n easy c o n s e q u e n c e of T h e o r e m 5.1 is as fol- [9] Abdulla S. Bin Shahna, On fuzzy compactness and fuzzylows. Lindel6fness, Bull. Cal. Math. Soc. 83 (1991) 146-150. [10] R. Srivastava and A.K. Srivastava, On fuzzy Hausdorff concepts, Fuzzy Sets and Systems 17 (1985) 67 71.Corollary 5.3. L e t X , Y, Z be f u z z y locally compact [11] C.K. Wong, Fuzzy points and local properties of fuzzyH a u s d o r f f spaces. Then the map topologies, J. Math. Anal. Appl. 46 (1974) 316-328. [12] C.K. Wong, Fuzzy topology: product and quotient the-S: Y x Ã— z r ~ Zx orems, J. Math. Anal. Appl. 45 (1974) 512 -521. [13] L.A. Zadeh. Fuzzy sets, 1nJorm. and Control 8 (1965)defined by S ( f g) = g o f is f u z z y continuous. 338 353.
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