Semicompactness in Fuzzy Topological Spaces

Transcription

1 BULLETIN of the Malaysan Mathematcal Scences Socety Bull. Malays. Math. Sc. Soc. (2) 28(2) (2005), Semcompactness n Fuzzy Topologcal Spaces R.P. Chakraborty, Anjana Bhattacharyya and M.N. Mukherjee Department of Pure Mathematcs, Unversty of Calcutta 35, Ballygunge Crcular Road, Kolkata 70009, Inda Abstract. The paper deals wth the concept of semcompactness n the generalzed settng of a fuzzy topologcal space. We acheve a number of characterzatons of a fuzzy semcompact space. The noton of semcompactness s further extended to arbtrary fuzzy sets. Such fuzzy sets are formulated n dfferent ways and a few pertnent propertes are dscussed. Fnally we compare semcompact fuzzy sets wth some of the exstng types of compact-lke fuzzy sets. We ultmately show that so far as the mutual relatonshps among dfferent exstng alled classes of fuzzy sets are concerned, the class of semcompact fuzzy sets occupes a natural poston n the herarchy Mathematcs Subject Classfcaton: Prmary 54A40; Secondary 54D99 Fuzzy semcompact space, sem-q-nbd, fuzzy sem- Key words and phrases: cluster pont.. Introducton Barrng paracompactness, there exsts n the lterature, a number of alled forms of compactness studed n a classcal topologcal space. Among these, the most wdely studed compact-lke coverng propertes are almost compactness or quas H-closedness of Porter and Thomas [8], near compactness of Sngal and Mathur [20], S-closedness of Thompson [22], s-closedness of Mao and Nor [8] and semcompactness of Dorsett [3]. The thorough nvestgatons and the applcatonal aspects of these coverng propertes have prompted topologsts to generalze these concepts (wth the excepton of semcompactness) to fuzzy settng. Malakar [9], n course of hs study of certan functons, ncdentally suggested the defnton of a fuzzy semcompact space. In [2] some of nterestng propertes of fuzzy semcompactness are nvestgated. Our ntenton here s to go nto some detals towards characterzatons of semcompactness for a fts. These characterzatons are effected wth the help of fuzzy nets, preflterbases and smlar other concepts, whch comprse the delberaton n the next secton. It s seen from the lterature that the process of generalzaton of dfferent coverng propertes, akn to compactness, to fuzzy perspectve was contnued further ahead n the form of extenson of such concepts to arbtrary fuzzy sets. Ths gave rse to the ntroducton and study of fuzzy compact, nearly compact, s-closed, S-closed, θ-rgd, Receved: Aprl 27, 2004; Revsed: March 9, 2005.

2 206 R.P. Chakraborty, A. Bhattacharyya and M.N. Mukherjee θ*-rgd, β-compact and almost compact sets respectvely abbrevated as FC-sets, FNC-sets, FsC-sets, FSC-sets, FθR-sets, Fθ*R-sets, FβC-sets and FAC-sets (see [4], [3], [2], [4],[3], [0], [6], [6] for detals). The nterrelatons among all these types of fuzzy sets are found to be as dsplayed by the followng dagram: FsC-set Fθ R-set FSC-set FC-set FNC-set FθR-set FAC-set where the concepts of fuzzy θ-rgdty (θ*-rgdty) and fuzzy almost compactness (S-closedness) concde f these are consdered for the whole fuzzy topologcal space, and no other mplcatons than those descrbed above, s true n general. Our aspraton, n Secton 3, would be to generalze the dea of fuzzy semcompactness to arbtrary fuzzy sets. Callng such fuzzy sets Fs*C-sets, we shall fnd some characterzatons of such sets along wth a few pertnent propertes. Our ultmate purpose of ntatng such fuzzy sets s fulflled by the way of establshng the followng mplcaton dagram, more balanced than the above one. (FβC-set ) Fs C-set FsC-set Fθ R-set FSC-set FC-set FNC-set FθR-set FAC-set We construct examples, to ths end, to show that a FsC-set or a FC-set need not be a Fs*C-set. In what follows, by a fts (X, τ) or smply by a fts X we shall mean a fuzzy topologcal space as defned by Chang [2]. The notatons cl A, nt A and A wll stand respectvely for the fuzzy closure [2], nteror [2] and complement [2] of a fuzzy set A n a fts X. The support of a fuzzy set A n X wll be denoted by supp A (.e., supp A = {x X : A(x) 0}). A fuzzy pont [9] n X wth the sngleton supp{x} X and the value α(0 < α ) wll be denoted by x α. The fuzzy sets n X takng on respectvely the constant values 0 and are denoted by 0 X and X respectvely. For two fuzzy sets A and B n X, we wrte A B f A(x) B(x), for each x X [23], whle we wrte A q B f A s quas-concdent (q-concdent, for short) wth B [9],.e., f A(x) + B(x) >, for some x X. The negaton of A q B s wrtten as AqB. A fuzzy set A n X s sad to be fuzzy regular open (semopen) f nt cl A = A (resp. U A cl U, for some fuzzy open set U) []. The complement A of a fuzzy semopen set A s called semclosed. The semclosure of a fuzzy set A n X, to be denoted by scl A, s the unon of all those fuzzy ponts x t such that for any fuzzy semopen set U wth U(x)+t >, there exsts y X wth U(y)+A(y) > [5]. A fuzzy semopen set U s called a sem-q-nbd of a fuzzy pont x α n a fts X f x α q U. A collecton F of fuzzy sets n a fts X s sad to form a preflterbase [7] n X f 0 X F and for any F, F 2 F, there exsts F 3 F such that F 3 F F 2. A collecton U of fuzzy sets n X s called a fuzzy cover of X f sup{u(x) : U U} = for all x X. 2. Fuzzy semcompact spaces We start wth the defnton of fuzzy semcompact spaces as suggested n [9].

3 Semcompactness n Fuzzy Topologcal Spaces 207 Defnton 2.. [9] A fts X s sad to be a fuzzy semcompact space f every fuzzy cover of X by fuzzy semopen sets (such a cover wll be called a fuzzy semopen cover of X) has a fnte subcover. A straghtforward consequence of the above defnton yelds the followng alternatve formulaton of a fuzzy semcompact space. Theorem 2.. A fts X s fuzzy semcompact ff each famly U of fuzzy semclosed sets n X wth fnte ntersecton property (.e., for every fnte subcollecton U 0 of U, U 0 0 X ) has a non-null ntersecton. In order to characterze fuzzy semcompact spaces by fuzzy nets and preflterbases we need the followng two defntons. Defnton 2.2. A fuzzy pont x α n a fts X s sad to be a fuzzy sem-cluster pont of a preflterbase B on X f x α scl B, for all B B. Defnton 2.3. [] A fuzzy pont x α n a fts X s sad to be a fuzzy sem-cluster pont of a fuzzy net {S n : n (D, )} [9] f for every sem-q-nbd W of x α and for each n D, there exsts m D wth m n such that S m q W. We now go on to fnd some characterzatons of fuzzy semcompact spaces. Theorem 2.2. A fts X s fuzzy semcompact ff every preflterbase on X has a fuzzy sem-cluster pont. Proof. Let X be fuzzy semcompact and let F = {F α : α Λ} be a preflterbase on X havng no fuzzy sem-cluster pont. Let x X. Correspondng to each n N (here and hereafter N denotes the set of natural numbers), there exsts a sem-q-nbd Ux n of the fuzzy pont x /n and an Fx n F such that Ux n qfx n. Snce Ux n (x) > /n, we have U x (x) =, where U x = {Ux n : n N}. Thus U = {Ux n : n N, x X} s a fuzzy semopen cover of X. Snce X s fuzzy semcompact, there exst fntely many members Ux n, Ux n 2,, U n k F Fx n Fx n2 2 F n k of U such that k = U x n = X. If F F such that, then F q X. Consequently, F = 0 X and ths contradcts the defnton of a preflterbase. Conversely, let every preflterbase have a fuzzy sem-cluster pont. We have to show that X s fuzzy semcompact. Let B = {F α : α Λ} be a famly of fuzzy semclosed sets havng fnte ntersecton property. The set of fnte ntersectons of members of B then forms a preflterbase F on X. So by the gven condton F has a fuzzy sem-cluster pont. Let x α be a fuzzy sem-cluster pont of F. So, x α α Λ scl F α = α Λ F α. Thus {F : F F} = 0 X. Hence by Theorem 2., X s fuzzy semcompact. Theorem 2.3. A fts X s fuzzy semcompact ff every fuzzy net n X has a fuzzy sem-cluster pont. Proof. Let X be a fuzzy semcompact space. If possble, let {S n : n (D, )}, where (D, ) s a drected set, be a fuzzy net n X whch has no fuzzy sem-cluster pont. For each fuzzy pont x α, there s a sem-q-nbd U xα of x α and an n Uxα D such that S m qu xα for all m D wth m n Uxα. Let U denote the collecton of all such U xα, where x α runs over all fuzzy ponts n X. Now the collecton V = { U xα : U xα U}

4 208 R.P. Chakraborty, A. Bhattacharyya and M.N. Mukherjee s a famly of fuzzy semclosed sets n X possessng fnte ntersecton property. In fact, let V 0 = { qu x α : =, 2,..., m} be a fnte subfamly of V. Then there exsts k D such that k n Ux,..., n Ux m α αm and so S p qu x α for =, 2,..., m and for all p k (p D),.e., S p m = U x xα = m = ( U x xα ) for all p k. Hence V 0 0 X. Snce X s fuzzy semcompact, by Theorem 2. there exsts a fuzzy pont y β n X such that y β { U xα : U xα U} = {U xα : U xα U}. Thus y β U xα, for all U xα U, and hence n partcular, y β U yβ.e., y β qu yβ. But by constructon, for each fuzzy pont x α there exsts a U xα U such that x α q U xα, and we arrve at a contradcton. To prove the converse, t suffces to prove, n vew of Theorem 2.2, that every preflterbase on X has a fuzzy sem-cluster pont. Let F be a preflterbase n X. As each F F s non-null, we choose a fuzzy pont x(f ) F. Let S = {x(f ) : F F}. Let a relaton be defned n F as follows: F α F β ff F α F β n X, for F α, F β F. Then (F, ) s a drected set. Now S s a fuzzy net wth the drected set (F, ) as doman. By hypothess the fuzzy net S has a fuzzy sem-cluster pont x t (0 < t ). Then for every sem-q-nbd W of x t and for each F F, there exsts G F wth G F such that x(g) q W. As x(g) G F, t then follows that F q W for each F F. Hence x t s a fuzzy sem-cluster pont of F. Defnton 2.4. A fuzzy net {S n : n (D, )}, where (D, ) s a drected set, s sad to be sem convergent to a fuzzy pont x α, f for every sem-q-nbd W of x α, there exsts m D such that S n q W, for all n m. Lemma 2.. A fuzzy pont x α s a fuzzy sem-cluster pont of a fuzzy net {S n : n (D, )}, where (D, ) s a drected set, n a fts X ff t has a fuzzy subnet whch fuzzy sem converges to x α. Proof. Let x α be a sem-cluster pont of the fuzzy net {S n : n (D, )}, wth the drected set (D, ) as the doman. Let W denote the collecton of all sem-q-nbds of x α. Now x α beng a fuzzy sem-cluster pont of the net {S n : n (D, )}, for each W W there exsts S n such that S n q W. Let C denote the set of all ordered pars (n, W ) wth the above property,.e., n D, W W and S n q W. Let us defne a relaton on C gven by (m, U)(n, V ) ff m n n D and U V. Then (C, ) s a drected set and t s easy to see that T : (C, ) (X, τ) gven by T (m, U) = S m s a fuzzy subnet of the gven fuzzy net. Let W be any sem-q-nbd of x α. Then there s an n D such that (n, W ) C and hence S n q W. Now, (m, U) C and (m, U)(n, W ) T (m, U) = S m q U and U W T (m, U)qW. Hence T sem converges to x α. The converse s clear. It now follows from Theorem 2.3 and Lemma 2. that Lemma 2.2. A fts X s fuzzy semcompact ff each fuzzy net n X has a fuzzy sem convergent subnet.

5 Semcompactness n Fuzzy Topologcal Spaces 209 Defnton 2.5. A fuzzy pont x α n a fts X s called a complete sem accumulaton pont of a fuzzy set A n X ff for each sem-q-nbd U of x α, supp A = {y X : A(y) + U(y) > }, where for a subset B of X, by B we mean, as usual, the cardnalty of B. Theorem 2.4. A necessary condton for a fts X to be fuzzy semcompact s that every fuzzy set A n X wth supp A N 0 (where N 0 denotes the cardnal number of the set of ntegers) has a complete sem accumulaton pont. Proof. Let A be a fuzzy set n a fuzzy semcompact space X such that supp A N 0, and f possble, suppose A has no complete sem accumulaton pont n X. Then for each x X and each n N, there s a sem-q-nbd Ux n of the fuzzy pont x /n (wth support x and value /n) such that (2.) {x X : A(x) + U n x (x) > } < supp A. Now, snce Ux n (x) + /n >, t follows that {Ux n : x X, n N} s a fuzzy cover of X by fuzzy semopen sets. As X s fuzzy semcompact, there exsts a fnte subset {x, x 2,, x m } of X and fntely many postve ntegers n, n 2,, n m such that m = U x n = X. Now, x supp A U n k =, for some k ( k m) U n k (x) + A(x) > x {y X : A(y) + U n k (y) > } = A Unk (say). As m = A U n, We have A Unk (2.2) supp A m =A Un. But A Unk < supp A by (2.) for =, 2,, m. Thus m = A Un = max A U n < supp A. m Hence, by (2.2) we get supp A m = A Un < supp A whch s a contradcton. Ths proves the result. Remark 2.. Notce that the converse of the theorem s false whch follows from the followng example. Example 2.. Consder a fuzzy set X wth the fuzzy topology τ = [0, ] X. Then the condton of the theorem s vacuously satsfed; but the fuzzy semopen cover τ\{ X } of X has no fnte subcover provng that the fts (X, τ) s not semcompact. 3. Fuzzy semcompact sets Before we ntroduce fuzzy semcompact sets, let us recall, to make the exposton clear, the defntons of certan exstng alled classes of fuzzy sets as follows. Defnton 3.. Let A be a fuzzy set n a fts X. A collecton U of fuzzy sets n X s sad to be a fuzzy cover of A [4] f sup{u(x) : U U} =, for all x supp A. If the members of U are fuzzy open (resp. regular open, semopen) n X, then U s called a fuzzy open (resp. regular open, semopen) cover of A. A fuzzy cover U of a fuzzy set A n X s sad to have a fnte (proxmate, sem-proxmate) subcover U 0 for A f U 0 s a fnte subfamly of U and U 0 A (resp. {cl U : U U 0 } A, {scl U : U U 0 } A).

6 20 R.P. Chakraborty, A. Bhattacharyya and M.N. Mukherjee Defnton 3.2. A fuzzy set A n a fts X s sad to be (a) a fuzzy compact set or smply a FC-set [4] f every fuzzy open cover of A has a fnte subcover for A, (b) a fuzzy nearly compact set, or a FNC-set [3] f every fuzzy regular open cover of A has a fnte subcover for A, (c) a fuzzy s-closed set (FsC-set [2]) f every fuzzy semopen cover of A has a sem-proxmate subcover for A, (d) a fuzzy almost compact set or a FAC-set [6] f every fuzzy open cover of A has a fnte proxmate subcover for A, (e) a fuzzy θ-rgd set or smply a FθR-set [3] f for every fuzzy open cover U of A, there exsts a fnte subfamly U 0 of U such that A nt cl( U 0 ). (f) A fuzzy θ -rgd or smply a Fθ R-set [0] f for every semopen cover U of A, there exsts a fnte subfamly U 0 of U such that A scl( {scl U : U U 0 }). We now set the followng defnton: Defnton 3.3. A fuzzy set A n a fts X s sad to be a fuzzy semcompact set (Fs*C-set, for short) f every fuzzy cover of A by fuzzy semopen sets of X has a fnte subcover for A. We would now proceed to obtan some characterzatons of the above type of fuzzy sets. Theorem 3.. For a fuzzy set A n a fts X, the followng are equvalent: (a) A s a Fs*C-set. (b) For every famly F of fuzzy semclosed sets n X wth {F : F F} A = 0 X, there exsts a fnte subcollecton F 0 of F such that F 0 qa. (c) If B s a preflterbase of fuzzy semclosed sets n X such that each element of B s q-concdent wth A, then ( B) A 0 X. Proof. (a) (b): Let A be a Fs*C-set n X and let F be a famly of fuzzy semclosed sets n X such that {F : F F} A = 0 X. Then, for every x supp A, nf{f (x) : F F} = 0, so that { F : F F} s a fuzzy semopen cover of A. Hence there exsts a fnte subcollecton F 0 of F such that A { F : F F 0 } A and consequently ( F 0 )qa. (b) (c): Obvous. (c) (a): If A s not a Fs*C-set n X, then there exsts a fuzzy semopen cover U of A whch has no fnte subcover for A. So for every fnte subcollecton U 0 of U there exsts x supp A such that sup{u(x) : U U 0 } < A(x),.e., nf{ U(x) : U U 0 } > A(x) 0. Thus f B = { U : U U}, then fnte ntersectons of members of B form a preflterbase F (say) of fuzzy semclosed sets n X for whch there s no member F of F such that F qa. In fact otherwse there exsts a fnte subset U 0 of U such that A { U : U U 0 } = {U : U U 0 }, contradctng our hypothess. By (c) we then have { U : U U} A 0 X and hence there exsts x supp A such that nf{ U(x) : U U} > 0,.e., sup{u(x) : U U} <, whch contradcts that U s a fuzzy cover of A. Theorem 3.2. A fuzzy set A n a fts X s a Fs*C-set ff whenever B s a preflterbase on X wth the property that for any F B and for any fuzzy semopen set U wth A U, F q U holds, then B has a fuzzy sem-cluster pont n A.

7 Semcompactness n Fuzzy Topologcal Spaces 2 Proof. Let A be a Fs*C-set. If possble, let B be a preflterbase wth the gven property, whch has no fuzzy sem-cluster pont n A. For each x supp A, there exsts a postve nteger m x such that /m x < A(x). For any postve nteger n m x, snce x /n A, x /n s not a fuzzy sem-cluster pont of B. Hence there s a sem-qnbd Vx n of x /n and a Bx n B such that Vx n qbx n. Snce Vx n (x) + /n >, we obtan sup n mx Vx n (x) =. The collecton U of all such Vx n for x supp A and n m x (> /A(x)), forms a fuzzy semopen cover of A such that for each Vx n U, there exsts Bx n B wth Vx n qbx n. Snce A s a fuzzy semcompact set n X, there exst a fnte number of members Vx n,, V n k of U such that A k = V x n = V (say). Let B B such that B k = B x. Then V s a fuzzy semopen set contanng A such that V qb. Conversely, let B be a preflterbase on X consstng of fuzzy semclosed sets such that {F : F B} A = 0 X. It then follows that B has no fuzzy sem-cluster pont n A. By hypothess, there exsts F B and there exsts a fuzzy semopen set U wth A U such that F qu. Then AqF. It then follows by Theorem 3.(c) that A s a Fs*C-set. The followng theorem s a generalzaton of the suffcency part of Theorem 2.2 for arbtrary fuzzy sets. Theorem 3.3. A fuzzy set A n a fts X s a Fs*C-set f every preflterbase n A has a fuzzy sem-cluster pont n A. Proof. If A s not a Fs*C-set, then there exsts a fuzzy semopen cover U of A such that for every fnte subcollecton U 0 of U A {U : U U 0 }. Correspondng to each U U, we defne a fuzzy set F U as follows : { mn{ U(x), A(x), A(x) U(x) } (x supp A) F U (x) = 0 (x supp A). For every fnte collecton {F U, F U2,, F Un } of members of F = {F U : U U}, we have sup U (x) < A(x), for some x supp A n so that mn[a(x) U (x),, A(x) U n (x)] > 0. Thus n = F U 0 X and consequently the famly B of fnte ntersectons of members of F s a preflterbase n A. Now for each fuzzy pont x α A, obvously there exsts U U such that x α q U. Snce F U qu, we have that B has no fuzzy sem-cluster pont n A. As to the converse of the last theorem, we have the followng result, the proof of whch s somewhat parallel to the necessty part of Theorem 2.2. Theorem 3.4. If A s a Fs*C-set n a fts X, then every preflterbase F n A, each of whose members s q-concdent wth A, has a fuzzy sem-cluster pont n A.

8 22 R.P. Chakraborty, A. Bhattacharyya and M.N. Mukherjee Proof. Let F be a preflterbase n A wth the gven property, such that F has no fuzzy sem-cluster pont n A. Consder any a supp A. Then for each postve nteger n wth n /A(a), as the fuzzy pont a /n ( A) s not a fuzzy sem-cluster pont of F, there exst a sem-q-nbd Ua n of a /n and an Fa n F such that Ua n qfa n. As Ua n (a) > /a, f we put U a = {Ua n : n s a natural number wth n /A(a)}, then U a (a) =. Hence U = {Ua n : a supp A, n /A(a)} s a fuzzy semopen cover of A. As A s a Fs*C-set, there exst fntely many members Ua n,, U n k a k of U such that k = U a n A. Now there s an F F wth F Fa n F n k a k. Then F q k = U a n so that F qa, and ths contradcts the stated condton on members of F. From the last two theorems, we obtan: Corollary 3.. A fuzzy set A n a fts X s a Fs*C-set ff every preflterbase F n A, each of whose members s q-concdent wth A, has a fuzzy sem-cluster pont n A. In the rest of ths secton, we derve a few elementary propertes concernng Fs*Csets. Theorem 3.5. In a fts, unon of fnte number of Fs*C-sets s a Fs*C-set. Proof. Clear. In order to look for the type of functons under whch fuzzy semcompactness remans nvarant, we recall the followng defnton. Defnton 3.4. [5] A functon f : X Y s sad to be a fuzzy rresolute functon f f (V ) s a fuzzy semopen set n X for every fuzzy semopen set V n Y. Theorem 3.6. If A s a Fs*C-set n a fts X and f : X Y s fuzzy rresolute then f(a) s a Fs*C-set n the fts Y. Proof. For each fuzzy semopen cover {V α : α Λ} of f(a) n Y, {f (V α ) : α Λ} s a fuzzy semopen cover of A n X. Hence Then A α Λ0 f (V α ), for some fnte subset Λ 0 of Λ. f(a) f ( α Λ0 f (V α ) ) ff ( α Λ0 V α ) α Λ0 V α. Thus f(a) s a Fs*C-set n Y. Corollary 3.2. If f : X Y s fuzzy rresolute and surjecton then Y s fuzzy semcompact whenever X s fuzzy semcompact. Defnton 3.5. [] A functon f : X Y s sad to be fuzzy semcontnuous f f (V ) s fuzzy semopen n X for every fuzzy open set V n Y. Theorem 3.7. If f : X Y s fuzzy semcontnuous then for any fuzzy set A n X, f(a) s a FC-set whenever A s a Fs*C-set. Proof. Clear.

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