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Journal of the Egyptian Mathematical Society (2014) 22, 70–82 Egyptian Mathematical Society Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems ORIGINAL ARTICLE Locally a-compact spaces based on continuous valued logic O.R Sayed * Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt Received 14 April 2013; revised 17 May 2013; accepted June 2013 Available online July 2013 KEYWORDS Łukasiewicz logic; Fuzzifying topology; Fuzzifying compactness; a-Compactness; Fuzzifying locally compactness; Locally a-compactness Abstract This paper is a continuation of [1] That is, it considers fuzzifying topologies, a special case of I-fuzzy topologies (bifuzzy topologies), introduced by Ying [2] It investigates topological notions defined by means of a-open sets when these are planted into the framework of Ying’s fuzzifying topological spaces (by Łukasiewicz logic in [0, 1]) Other characterizations of fuzzifying acompactness are given, including characterizations in terms of nets and a-subbases Several characterizations of locally a-compactness in the framework of fuzzifying topology are introduced and the mapping theorems are obtained 2000 MATHEMATICS SUBJECT CLASSIFICATION: 54A40; 54B10; 54D30 ª 2013 Production and hosting by Elsevier B.V on behalf of Egyptian Mathematical Society Introduction and preliminaries In the last few years fuzzy topology, as an important research field in fuzzy set theory, has been developed into a quite mature discipline [3–8] In contrast to classical topology, fuzzy topology is endowed with richer structure, to a certain extent, which is manifested with different ways to generalize certain classical concepts So far, according to Ref [4], the kind of topologies defined by Chang [9] and Goguen [10] is called the topologies of fuzzy subsets, and further is naturally called L -topological spaces if a lattice L of membership values has been chosen Loosely speaking, a topology of fuzzy subsets (resp an L -topological space) is a family s of fuzzy subsets * Tel.: +20 1115242205 E-mail addresses: o_sayed@aun.edu.eg, o_r_sayed@yahoo.com Peer review under responsibility of Egyptian Mathematical Society Production and hosting by Elsevier (resp L -fuzzy subsets) of nonempty set X, and s satisfies the basic conditions of classical topologies [11] On the other hand, Hoăhle in [12] proposed the terminology L-fuzzy topology to be an L-valued mapping on the traditional powerset P(X) of X The authors in [6,7,13,14] defined an L-fuzzy topology to be an L-valued mapping on the L-powerset LX of X In 1952, Rosser and Turquette [15] proposed emphatically the following problem: If there are many-valued theories beyond the level of predicates calculus, then what are the detail of such theories ? As an attempt to give a partial answer to this problem in the case of point set topology, Ying in 1991-1993 [2,16,17] used a semantical method of continuous-valued logic to develop systematically fuzzifying topology Briefly speaking, a fuzzifying topology on a set X assigns each crisp subset of X to a certain degree of being open, other than being definitely open or not Roughly speaking, the semantical analysis approach transforms formal statements of interest, which are usually expressed as implication formulas in logical language, into some inequalities in the truth value set by truth valuation rules, and then these inequalities are demonstrated in an algebraic way and the semantic validity of conclusions is thus 1110-256X ª 2013 Production and hosting by Elsevier B.V on behalf of Egyptian Mathematical Society http://dx.doi.org/10.1016/j.joems.2013.06.003 Locally a-compact spaces based on continuous valued logic established There are already more than 100 papers in fuzzifying topology published in the last two decades, I guess But only a few papers can properly use the semantic method introduced in the original papers of Ying, which I strongly believe, can provide more delicate characterization of fuzzifying topological structure So far, there has been significant research on fuzzifying topologies [18–24] For example, Ying [22] introduced the concepts of compactness and established a generalization of Tychonoff’s theorem in the framework of fuzzifying topology In [24] the concept of local compactness in fuzzifying topology is introduced and some of its properties are established In [18] the concepts of fuzzifying a-open set and fuzzifying a-continuity were introduced and studied Also, Sayed [21] introduced some concepts of fuzzifying a-separation axioms and clarified the relations of these axioms with each other as well as the relations with other fuzzifying separation axioms Quite recently, Sayed in [1] used the finite intersection property to give a characterization of fuzzifying a-compact spaces In classical topology , a-compact spaces and locally a-compact spaces have been studied in [25,26] In this paper, the concepts of a-base and a-subbase of fuzzifying a-topology are introduced Other characterizations of fuzzifying a-compactness are given, including characterizations in terms of nets and asubbase Several characterizations of locally a-compactness in the framework of fuzzifying topology are introduced and the mapping theorems are obtained Thus we fill a gap in the existing literature on fuzzifying topology We use the terminologies and notations in [1,2,16–18,21,22,24] without any explanation We note that the set of truth values is the unit interval and we often not distinguish the connectives and their truth value functions and state strictly our results on formalization as Ying does We will use the symbol instead of the second ‘‘AND’’ operation ^ as dot is hardly visible This mean that Á [a] [u fi w] () [a] [u] [w] All of the contributions in general topology in this paper which are not referenced may be original We now give some definitions and results which are useful in the rest of the present paper The family of all fuzzifying aopen sets [18], denoted by sa IðPðXÞÞ, is defined as A sa :¼ 8xðx A ! x IntClIntAịịịị; i: e:; sa Aị ^ IntClIntAịịịxị: ẳ x2A The family of all fuzzifying a-closed sets [18], denoted by Fa IPXịị, is dened as A Fa :ẳ X À A sa The fuzzifying a-neighborhood system of a point x X [18] is denoted by Á W XÀ Nax or Nax IðPðXÞÞ and defined as Nax Aị ẳ sa Bị x2B # A The fuzzifying a-closure of a set A ˝ X [18], denoted by Cla IXị, is dened as Cla Aịxị ẳ À Nax ðX À AÞ If (X, s) is a fuzzifying topological space and N(X) is the class of all nets in X, then the binary fuzzy predicates a ; /a IðNðXÞ Â XÞ [23] are defined as X X ~ ; S/a x :¼ 8A A Nax ! S@A ~ ; S.a x :¼ 8A A Nax ! S&A where ‘‘S.ax’’, ’’S µ ax’’ stand for ‘‘S a-converges to x’’ , ‘‘x is ~ ‘‘@’’ ~ are an a-accumulation point of S’’, respectively; and ‘‘&’’, the binary crisp predicates ‘‘almost in ‘‘,’’often in’’, respectively The degree to which x is an a-adherence point of S is adhaS(x) = [S µ ax] If (X, s) and (Y, r) are two fuzzifying topological spaces and f YX, the unary fuzzy predicates 71 Ca ; Ia IðYX Þ; called fuzzifying a-continuity [18], fuzzifying a-irresoluteness [1], are given as Ca(f) :¼ "B(B r fi fÀ1(B) sa), Ia(f):¼"B(B fi fÀ1(B) sa), respectively Let X be the class of all fuzzifying topological spaces A unary fuzzy predicate Ta2 IðXÞ, called fuzzifying a-Hausdorffness [21], is given as follows: Ta2 ðX; sÞ ¼ 8x8yððx X ^ y X ^ x–yÞ ! 9B9CðB Nax ^ C Nay ^ B \ C /ÞÞ: A unary fuzzy predicate C IðXÞ, called fuzzifying compactness [22], is given as follows: CðX; sị :ẳ 8RịK R; Xị ! 9}ị} Rị ^ Kð}; AÞ FFð}ÞÞÞ and if A ˝ X, then C(A):¼C(A, s/A) For K, K (resp and FF) see [16, Definition 4.4] (resp [16, Theorem 4.3] and [22, Definition 1.1 and Lemma 1.1]) A unary fuzzy predicate fI IðIðPðXÞÞÞ, called fuzzy finite intersection property [22], is given as fIRị :ẳ 8}} Rị ^ FF}ị ! 9x8BB } ! x BÞÞ: A fuzzifying topological space (X, s) is said to be fuzzifying atopological space [1] if sa(A \ B) P sa(A) Ù sa(B) A unary fuzzy predicate LC IðXÞ, called fuzzifying locally compactness [24], is given as follows: (X, s) LC:¼("x)($B)((x Int(B) C(B, s/B)) Fuzzifying a-base and a-subbase Definition 2.1 Let (X, s) be a fuzzifying topological space and ba ˝ sa Then ba is called an a-base of sa if ba fulfils the condition: X A Nax ! 9BððB ba Þ ^ ðx B # AÞÞ: Theorem 2.1 bais an a-base of sa if and only if sa ẳ b[ị a , where b[ị a Aị ¼ _ S ^ B ¼A k2K k ba ðBk Þ: k2K Proof Suppose that ba is an a-base of sa If [ Bk ¼ A; k2K then from Theorem 3.1 (1) (b) in [18], ! [ ^ ^ sa Aị ẳ sa Bk P sa Bk ị P ba ðBk Þ: k2K k2K Consequently, _ ^ sa ðAÞ P ba Bk ị: [ k2K Bk ẳA k2K To prove that _ sa Aị S B ẳA k2K k ^ ba ðBk Þ; k2K we first prove ^ _ sa Aị ẳ sa Bị: x2Ax2B # A k2K 72 O.R Sayed (Indeed, assume cx = {B:x B ˝ A} Then for any Y [ f2 c ; fðxÞ ¼ A; x2A x Theorem 2.2 Let ba IðPðXÞÞ Then ba is an a-base for some fuzzifying a-topology sa if and only if it has the following properties: x2A (1) b[ị a X ị ẳ 1; (2) A ba Þ ^ ðB ba Þ ^ ðx A \ BÞ ! 9CððC and furthermore [ sa Aị ẳ sa ! fxị x2A ẳ ^ P x2A ^ _ _ sa ðfðxÞÞ P f2 Q x2A ^ sa ðfðxÞÞ sa ðBÞ: Proof If ba is an a-base for some fuzzifying a-topology sa, [ị then sa Xị ẳ b[ị a Xị Clearly, ba Xị ẳ In addition, if x A \ B, then sa ðBÞ: ba ðAÞ ^ ba ðBÞ sa ðAÞ ^ sa ðBÞ sa ðA \ BÞ Nax ðA \ BÞ _ ba ðCÞ: x2Ax2B # A Also sa ðAÞ ^ _ X x2Ax2B # A Therefore ^ _ sa Aị ẳ sa Bịị: x2C # A\B x2Ax2B # A Now, since _ X Nax ðAÞ ba ðBÞ; sa Aị ẳ x2B # A ba ị ^ ðx C # A \ BÞ: cx x2A ^ _ sa Bị ẳ x2Ax2B # A ^ _ _ ba Bị ẳ x2Ax2B # A ^ Q f2 x2A ^ X Nax ðAÞ Conversely, if ba satisfies (1) and (2), then we have sa is a fuzzifying a-topology In fact, sa(X) = For any {Ak:k K} ˝ P(X), we set ( ) [ ck ¼ fBdk : dk Kk g : Bdk ¼ Ak : dk 2Kk x2A ba ðfðxÞÞ: cx x2A Then for any Y [ [ [ f2 c; Bdk ¼ Ak : k2K k k2KBd 2fðkÞ k Then sa ðAÞ _ ^ [ Bk ¼A Therefore ! [ sa Ak ¼ ba Bk ị: k2K k2K k2K Therefore sa Aị ẳ _ ^ [ Bk ¼A P k2K P In the other side, we assume _ ^ [ Bk ¼A x2B # A [ Bk ¼ B; k2K _ ba ðBk Þ ba ðBk Þ ba ðBÞ: x2B # A k2K Therefore X Nax Aị ẳ _ sa Bị ¼ x2B # A ^ ^ c k2K k ba ðBdk Þ k2KBd 2fðkÞ k _ ^ ba ðBdk Þ ¼ ^ sa ðAk Þ: k2K [ k1 2K1 Bk1 ¼ A; [ Bk2 ¼ B k2 2K2 and for any k1 K1 ; ba ðBk1 Þ > t, for any k2 K2 ; ba ðBk2 Þ > t Now, for any x A \ B, there exists k1x K1, k2x K2 such that x Bk1x \ Bk2x From the assumption, we know that _ t < ba ðBk1x Þ ^ ba ðBk2x Þ ba ðCÞ x2C # Bk1x \Bk2x then there exists k K such that x Bk and ^ ^ Ak ba ðBd Þ d2K k2K _ Q [ Finally, we need to prove that sa(A \ B) P sa(A) Ù sa(B) If sa(A) > t, sa(B) > t, then there exists fBk1 : k1 K1 g; fBk2 : k2 K2 g such that ba ðBk Þ k2K and we will show that ba is an a-base of sa, i.e., for any _ X A # X; Nax ðAÞ ba ðBÞ: x B # A; Bd ¼ ^ k2KfBd :dk 2Kk g2ck dk 2Kk k k2K Indeed, if [ f2 k2K sa ðAÞ ¼ _ d2K ba ðBk Þ k2K _ x2B # A _ _ ^ [ x2B # A k2K ba Bị: k2K Bk ẳB ba Bk ị and furthermore, there exists Cx such that x Cx # Bk1x \ Bk2x # A \ B; ba ðCx Þ > t: S Since Cx ¼ A \ B, we have x2A\B ^ _ ^ t6 ba ðCx Þ ba Bk ị ẳ sa A \ Bị: [ x2A\B k2K Bk ¼A\B k2K Now, let sa(A) Ù sa(B) = k For any natural number n, we have sa ðAÞ > k À 1n ; sa ðBÞ > k À 1n and so sa ðA \ BÞ P k À 1n Therefore sa(A \ B) P k = sa(A) Ù sa(B) h Locally a-compact spaces based on continuous valued logic 73 Recall that if (X, s) is a topological space and sa is the collection of all a-open sets in X, then an a-subbase of sa is a collection S of a-open sets such that every a-open set is the union of sets that are finite intersections of elements from S Therefore we have the following definition Definition 2.2 ua IðPðXÞÞ is called an a-subbase of sa if uea is an a-base of sa, where uea \ k2K ! Bk ¼ _ T k2K ^ b5 :¼ ð8RÞðR IðPðXÞÞ ^ R # Fa fIðRÞ ! 9x8AðA R ! x AÞÞ: Then `(X, s) Ca M bi i = 1, 2, , Proof (1) Since ua # sa ; ½R # ua ½R # sa for any R IP X ịị Then ẵK ua R; X ị ½K a ðR; X Þ Therefore Ca(X, s) ) [b1] ^ ( _ a (2) ẵb2 ẳ ẵS x : S is a universal net in X : ua Bk ị; fBk : k KgbPXị; Bk ẳAk2K x2X with ‘‘b’’ standing for ‘‘a finite subset of’’ Theorem 2.3 ua IðPðXÞÞ is an a-subbase of some fuzzifying a-topology if and only if u[ị a Xị ẳ Proof We only demonstrate that uea satisfies the second condition of Theorem 2.2, and others are obvious In fact B uea Aị ^ uea Bị ẳ @ _ T k1 2K1 ¼ k1 2K1 ^ Bk1 ¼Ak1 2K1 _ T Bk1 ¼A _ T k2K Bk ¼A\B C B ua ðBk1 ÞA ^ @ T k2 2K2 _ T k2 2K2 _ ^ Bk2 ¼Bk2 2K2 ! ^ Bk2 ẳB ^ ua Bk1 ị ^ k1 2K1 C ua ðBk2 ÞA ^ ! ua ðBk2 Þ k2 2K2 ! ua ðBk Þ _ uea ðAÞ ^ uea ðBÞ uea ðA \ BÞ uea ðCÞ: 1À Naxi ðAÞ À à x2C # A\B Fuzzifying a-compact spaces ~A Så ¼ Therefore [b2] = P [b1] (2.2) In general, to prove that [b1] [b2] we prove that for any k [0, 1], if [b2] < k, then [b1] < k Assume for any k [0, 1], [b W2] À k Since ua is an a-subbase of sa ; ua is an a-base ofW sa and from Definition 2.1, we have uea ðBÞ P Nax ðAÞ > À k, i.e., there exists B ˝ A such Nax ðAÞ x2B # A that x B ˝ A and ( ) _ \ mink2K ua Bk ị : Bk ẳ B; Bk # X; k K ẳ uea Bị k2K Denition 3.1 A binary fuzzy predicate Ka IðIðPðXÞÞ PðXÞÞ, called fuzzifying a-open covering [1], is given as Ka R; Aị :ẳ KR; AÞ ðR # sa Þ A unary fuzzy predicate Ca IðXÞ, called fuzzifying a-compactness [1], is given as follows: X; sị Ca :ẳ 8RịKa R; Xị!9}ị} RÞ ^ Kð}; XÞ FFð}ÞÞÞ and if A ˝ X, then Ca(A):¼Ca(A, s/A) It is obvious that Ca(X, s):¼C(X, sa) and K ðR; AÞÀ!Ka ðR; AÞ Theorem 3.1 Let (X, s) be a fuzzifying topological space, ua be an a-subbase of sa, and b1 :ẳ 8RịKua R; Xị ! 9}ðð} RÞ ^ Kð}; XÞ FFð}ÞÞÞ; where Kua R; Xị :ẳ KR; Xị R # ua Þ; b2 :¼ ð8SÞððS is a universal net in XÞ ! 9xx Xị ^ S.a xịị; b3 :ẳ 8SịS NðXÞ ! ð9TÞð9xÞððT < SÞ ^ ðx XÞ ^ ðT.a xÞÞ; where ‘‘T < S’’ stands for ‘‘T is a subnet of S; b4 :ẳ 8SịS NXị ! :ðadha S /ÞÞ; > À k; where K is finite Therefore there T exists a finite set K and Bk ˝ X(k K) such that Bk ¼ B and for any k2K ~ A and K is finite, there exists k K, ua(Bk) > À k Since S å W ~ ua ðBkðxÞ Þ If k(x) K such that S å BkðxÞ We set R Bkxị ị ẳ x2X } R , then for any d > 0,}d ˝ {Bk(x):x X} Consequently, ~ B and S&B ~ c because S is a universal net If for any B }d ; S å [FF(})] = À inf{d [0, 1]:F(}d)} = t, then for any n w (the non-negative integer), inffd ẵ0; : F}d ịg < t ỵ 1n, and there exists d < t ỵ 1n such that F(}d) If d = 0, then P(X) = }d is finite and it is proved in (2.1) If d > 0, then ~ c Since F(}d), we have for T any B }d ; S&B S&~ fBc : B }d g–/ i.e., ¨}d „ Xand there exist x X such that for any B }d, x R B Therefore, if x B, then B R }d, i.e., ^_ _ }Bị < d; K}; Xị ẳ }Bị }Bị d < t ỵ : n x2Xx2B x 2B Let n fi We obtain K(}, X) À t and [K(}, X) FF(})] = In addition, ẵKua R ; Xị P k In fact, ẵR # ua ẳ and 74 O.R Sayed ^_ ẵKR ; Xị ẳ R Bị P x2Xx2B ^ R ðBkðxÞ Þ P x2X ^ ua Bkxị ị x2X ẵb4 ẵ:adha S /ị ¼ Assume ½R # Fa ¼ l Then for any B PXị; RBị ỵ Fa Bị À l, and because x Bk(x) Now, we have ẵb1 ẳ 8RịKua R; Xị ! 9}} Rị ^ Kð}; XÞ FFð}ÞÞÞ Kua ðR ; XÞ ! 9}ðð} R Þ ^ Kð}; XÞ FFð}ÞÞ _ ẵK}; Xị FF}ịị k: ẳ min1; Kua R ; Xị ỵ }6R ~A S 6@ x2XxRA x X; ^À ^ Á À Nax Aị l k ỵ RðAÞÞ; ~A S6@ xRA i.e., _ RðAÞ À l k ỵ ^ ~A T t, then xRA there exists A such that x R A and RðA Þ > t W a Case t À k, then t À l À k ỵ N x Aị $ S6@ A Case t > À k Here we set d ¼ A Rd ; A #d In addition, t ỵ kị and have t < RA ị ỵ Fa A ị l; t ỵ l Fa A ị ẳ sa Ac : Since A #d,$ we know that SB A, i.e., SB R Ac when B ˝ A and S 6@ Ac Therefore, 2lkỵ _ Nax Aị P l k ỵ Nax Ac P l k ~A S6@ ỵ sa Ac P t þ ð1 À kÞ P t: The above theorem is a generalization of the following corollary S2NðXÞx2X ! ð1 À ½S/a xÞ Nax ðAÞ By noticing that t is arbitrary, we have completed the proof (6) To prove that [b5] = [(X, s) Ca] see [1] Theorem 3.3 h ~A S6@ S2NðXÞx2X _ ~A S6@ xRA ~ Ag as follows Suppose T = S ~ Ag # fA : T å fA : S 6@ ~ A, then there exists m D such that S(n) R A K If S 6@ when n P m, where P directs the domain D of S ~ A If not, then there exists Now, we will show that T å p E such that T(q) A when q P p, where P directs the domain E of T Moreover, there exists n1 E such that K(n1) P m because T < S, and there exists n2 E such that n2 P n1, p because (E, P) is directed So, K(n2) P K(n1) P m, SK(n2) R A and SK(n2) = ~ Ag # T(n2) A They are contrary Hence fA : S 6@ ~ Ag Therefore fA : T _ ^ ẵ9TT < Sị ^ T.a xịị ẳ Nax Aị ẳ ẵR # Fa fIðRÞ ! ð9xÞð8AÞððA RÞ ! x Aị _^ ẳ min1; l k þ ð1 À RðAÞÞÞ: Therefore, it suffices to show that for any By noticing that k is arbitrary, we have [b1] [b2] (3) It is immediate that [b2] [b3] (4) To prove that [b3] [b4], first we prove that [$T ((T < S) Ù (T.ax))] [S ax], where ẵ9T T < Sị W V a N ax Aị ^T xịị ẳ and ẵS/a x ẳ ~A T À k, if A1 ; ; W An Rd ; A1 \ A2 \ \ An –/ In fact, we set }Ai ị ẳ niẳ1 RAi ị Then } R and FF(}) = By putting e = k + d À > 0, we obtain k e < k ẵFF}ị ! 9xị8BịB } ! x Bị _^ ẳ }Bịị: (a) X is an a-compact space (b) Every cover of X by members of an a-subbase of sa has a finite subcover (c) Every universal net in X a-converges to a point in X (d) Each net in X has a subnet that a-converges to some point in X (e) Each net in X has an a-adherent point (f) Each family of a-closed sets in X that has the finite intersection property has a non-void intersection x2XxRB There exists x X such that k À e < V ð1 À }ðBÞÞ; x R B im- x RB plies }(B) < À k + e = d and x \ }d = A1 \ A2 \ \ An Now, we set #d ¼ fA1 \ A2 \ \ An : n N; A1 ; ; An Rd g and S:#d fi X, B ´ xB B, B #d and know that (#d, ˝) is a directed set and S is a net in X Therefore Definition 3.2 Let {(X Qs, ss):s S} be a family of fuzzifying topological spaces, Xs be the cartesian product of Ès2SÀ1 É , where {Xs:s S} and u ¼ p ðU Þ : s S; U PðX Þ s s s s Q pt : s2S Xs ! Xt ðt SÞ is a projection For U ˝ u, S(U) stands Àfor of indices of elements in U The a-base Q theÁ set Q ba I s2S Xs of s2S ðsa Þs is defined as Locally a-compact spaces based on continuous valued logic \ V ba :ẳ 9Uị Ubu ^ U ẳ V ! 8sðs SðUÞ ! Vs ðsa Þs Þ; i:e:; _ ^ ba ðVÞ ¼ ðsa Þs ðVs Þ: T s2SðUÞ Ubu; U¼V Definition 3.3 Let (X, s) and (Y, r) be two fuzzifying topological space A unary fuzzy predicate Oa I(YX), is called fuzzifying a-openness, is given as follows: Oa(f) :¼ "U(U sa fi f(U) ra) Intuitively, the degree to V which f is a-open is ẵOa fị ẳ min1; sa Uịỵ U#X ðfðUÞÞÞ Lemma 3.1 Let (X, s) and (Y, r) be two fuzzifying topological space For any f YX, Oa fị :ẳ 8BB bXa ! fBị Þ, where bXa is an a-base of sa  Proof Clearly, ẵOa fị 8U U bXa ! fðUÞ Conversely, for any U ˝ X, we are going to prove  À Áà min1; sa Uị ỵ fUịịị P 8V V bXa ! fðVÞ : If sa(U) ra(f(U)), it is hold clearly Now assume S S sa(U) > raS (f(U)) If R # PðXÞ with R ¼ U, then V2R fðVÞ ¼ fð RÞ ¼ fðUÞ Therefore _ ^ X sa Uị fUịị ẳ ba Vị S V2R R # PXị; RẳU _ ^ S }ẳfUịW2} } # PYị; _ S R # PðXÞ; ^ R # PðXÞ; S S R # PðXÞ; s2SðUk Þ pt(U) = / and (sa)t(pt(U)) = Therefore ÀQ Á ðsa Þt ðpt ðU ÞÞ P s2S ðsa Þs ðU Þ T À1 ps ðV s Þ ¼ Bk , (2) If there exists k K, such that /– s2SðUk Þ If t R SðUk Þ, i.e., t S À SðUk Þ; pt ðBk Þ ¼ X t Therefore ðsa Þt ðpt ðBk ÞÞ ¼ ðsa Þt ðX t Þ ¼ (ii) If t SUk ị, then pt Bk ị ẳ V t # X t Thus 00 11 [ [ pt Uị ẳ pt @@ Bk A [ @ Bk AA (i) ^À ¼@ _ S ^ À 1; À bXa ðVÞ Then it suffices to show that for any U P ðsa Þt ðpt ðUÞÞ P s2S Assume Y ðs a Þs ðUÞ ¼ s2S ÀQ s2S Xs Á ^ s2S C Nax ðUÞA Xs Ca ðXt ; st Þ: _ ^ Indeed, if ÀQ , we have U2P s2S Xs Á B @Ca ðU; s=UÞ ^ _ Q x2 s2S C Nax ðUÞA > l > 0; Xs ÀQ Á then there exists U P s2S Xs such that Ca(U, s /U) > l and Á W ÀQ W Nax Uị > l, where Nax Uị ẳ s2S ðsa Þs ðVÞ Q x2V # U ðsa Þs ðUÞ: _ x2 Xs _ ^ _ s2S Xs ^ Ca ðU; s=UÞ ^ 9xðx Inta ðUÞÞ TbSt2SÀT Y 1; À ðsa Þs ðUÞ þ ðsa Þt ðpt ðUÞÞ : s2S Y Y Proof It suffices to show that _ _ B @Ca ðU; s=UÞ ^ Q ÀQ Á Xs pt ðBk ÞA ¼ Vt [ Xt ¼ Xt : tRSðUk Þ ! 9TðTbS ^ 8tðt S À T ^ Ca ðXt ; st ÞÞÞ: (1) For any t S, we have s2S pt ðBk ÞA [ @ From Lemma 3.1 in [17] we have `("s)(s S fi p s C) Furthermore, for any two fuzzifying topological spaces (X, s) and (Y, r) and f YX, we have C(f) Ca(f) (Theorem 6.3 (3) in [18]) Therefore `("s)(s S fi ps Ca) h 9UðU # U2P U2P [ Theorem 3.2 Let {(Xs, ss):s S} be the family of fuzzifying topological spaces, then à Proof Á tRSðUk Þ s2S Lemma 3.2 For any family {(Xs, ss) : s S} of fuzzifying topological spaces (1) `("s)(s S fi ps Oa); and (2) `("s)(s S fi ps Ca) ^ Hence (sa)t(pt(U)) = (sa)t(Xt) = or (sa)t(pt(U)) = (sa)t(Vt) > l ÀQ Á Therefore ðsa Þt ðpt ðUÞÞ P s2S sa ịs Uị Thus Oa(pt) = RẳUV2R ỵra fVịịị P 8V V bXa ! fðVÞ : t2SðUk Þ ðfðVÞÞ Á bXa ðVÞ À ðfðVÞÞ ; R # PðXÞ; ÀQ [ (2) RẳUV2R min1; sa Uị ỵ fUịịị P Oa pt ị ẳ t2SUk ị RẳUV2R _ k2K more, for any k K, there exists Uk b u such that \Uk = Bk T and ps Vs ị ẳ Bk , where for any s S(Uk) we have s2SðUk Þ ! S T (sa)s(Vs) > l Thus pt Uị ẳ pt ps ðVs Þ k2K s2SðUk Þ T À1 (1) If for any k K; ps ðV s Þ ¼ /, then U = /, bXa ðVÞ ^ É where Uk ẳ p1 s Vs ị : s SðUk Þ ðk KÞ.Hence there exists Q S fBk : k Kg # P s2S Xs such that Bk ẳ U and further- RẳUV2R _ Wị 75 È x2 ðsa Þs ðVs Þ > l; [k2K Bk ẳUk2KUk bu;\Uk ẳBk s2SUk ị s2S Xs Furthermore, there exists V such that x V ˝ U and ÀQ Á Q s2S ðsa Þs ðVÞ > l Since ba is an a-base of s2S ðsa Þs ; 76 O.R Sayed Y sa ịs Vị ẳ s2S _ ^ _ ba Bk ị ẳ [k2K Bk ẳVk2K ^ _ ^ Lemma 3.4 For any fuzzifying a-topological ðX; sÞ; Ta2 ðX; sÞ Ca ðX; sÞ ! Ta4 ðX; sÞ ðsa Þs ðVs Þ > l; [k2K Bk ¼Vk2KUk bu;\Uk ¼Bk s2SðUk Þ where È É Uk ¼ pÀ1 s ðVs Þ : s SðUk Þ ðk KÞ: For the definition of Ta4 ðX; sÞ see [21] ÀQ Á Hence there exists fBk : k Kg # P s2S Xs such that [k2KBk = V Furthermore, for any k K, there exists Uk b u such that \Uk = Bk and for any s S(Uk), we have (sa)s(Vs) > l Since x V, there exists Bkx such that x Bkx # V # U Hence there exists Ukx bu such that \Ukx ẳ Bkx and \ p1 s Vs ị ẳ Bkx # Y s2S Xs  à Proof If Ta2 X; sị CaX; sị ẳ 0, then the result holds a Now, suppose that T2 ðX; sÞ Ca ðX; sÞ > k > Then Ta2 ðX; sị ỵ Ca X; sị > k > Therefore from Theorem 4.6 [1], Ta2 ðX; sÞ Ca Aị ^ Ca Bịị ^ A \ B ẳ /Þws Ta2 ðX; sÞ ! ð9UÞð9VÞððU sa Þ ^ ðV s2SðUk Þ sa Þ ^ ðA # UÞ ^ B # Vị ^ A \ B ẳ /ịị: and for any s S(Uk), we have (sa)s(Vs) > l By \ p1 s Vs ị ẳ Bkx ; Then for any A, B ˝ X, A \ B = /, s2SðUk Þ Ta2 ðX; sÞ ðCa ðAÞ ^ Ca ðBÞÞ we have Pd ðBkx Þ ¼ Vd # Xd , if d SðUkx Þ; Pd ðBkx Þ ¼ Xd , if d S À SðUkx Þ Since Bkx # U, for any d S À SðUkx Þ, we have Pd ðUÞ Pd Bkx ị ẳ Xd and Pd(U) = Xd On the other hand, since for any s S and Us PðXs Þ; Y À Á pÀ1 ðs a Þt s ðUs Þ P ðsa Þs ðUs Þ; t2S ^ Ca ðXd ; sd Þ P Ca ðU; s=UÞ > l: From Theorem 4.1(1) A Fa ! Ca ðAÞ Then Corollary 3.2 If there exists a coordinate a-neighborhood a-compact subset U of some point x X of the product space, then all except a finite number of coordinate spaces are a-compact Lemma 3.3 For any fuzzifying topological ðX; sÞ; A # X; Ta2 ðX; sÞ ! Ta2 A; s=Aị ^ _ x;y2X;xyU;V2PXị;U\Vẳ/ ^ _ Nax ðUÞ; Nay ðVÞ ^ _ A Ca X; sị ẵCa Aị ^ Ca Bị: c c Thus Ca(X, s) À [Ca(A) Ù Ca(B)] W À À [sa(A ) Ù sa(B )] So, À ẵsa Ac ị ^ sa Bc ị ỵ minsa Uị; sa Vịị > k, U\Vẳ/;A # U;B # V i.e., Ta4 X; sị ẳ ^ A\Bẳ/ min1; ẵsa ðAc Þ ^ sa ðBc Þ _ minðsa ðUÞ; sa ðVÞÞÞ > k: à The above lemma is a generalization of the following corollary A Nax ðU0 Þ; Nay V0 ị ẳ Ta2 A; s=A ị; BẳV\A have U\Vẳ/;A # U;B # V x;y2A;xyU0 \V0 ẳ/;U0 ;V0 2PAị W A where Nax Uị ẳ sa =ACị W x2C # U sa ðVÞ h we ^ ðCa ðX; sÞ sa Bc ịị ỵ A A Nax U \ Aị; Nay V \ Aị x;y2X;xyU\Aị\V\Aịẳ/ [1] ðCa ðX; sÞ sa ðAc ÞÞ space Proof in ^ Ca X; sị ỵ sa Bc ị 1Þ Ã The above theorem is a generalization of the following corollary minðsa ðUÞ; sa ðVÞÞ: Ca ðX; sÞ ỵ ẵsa Ac ị ^ sa Bc ị ẳ Ca X; sị ỵ sa Ac ị 1ị d2SSUk ị a T2 X; sị ẳ _ Ta2 ðX; sÞ Ca ðAÞ ^ Ca ðBÞ ! U\V¼/;A # U;B # V Furthermore, since by Theorem 4.4 [1], we have `Ca(X, s) Ia(f) fi Ca(f(X)), then Ca(U, s/U) = Ca(U, s/ U) Ia(pd) Ca(Pd(U), sd) = Ca(Xd, sd) for each d SÀ SðUk Þ Therefore, TbSt2SÀT or equivalently A; B # X; A \ B ẳ /; ẵCa Aị ^ Ca Bị _ ỵ minsa Uị; sa Vịị ỵ Ca X; sÞ À > k: Us 2PðXs Þ Ca ðXt ; st ị P minsa Uị; sa Vịị U\Vẳ/;A # U;B # V Hence for any Y À Á 1; sa ịs Us ị ỵ t2S sa ịt p1 ẳ 1: s Us ị ^ s S; Ia ps ị ẳ _ U\Vẳ/;A # U;B # V we have, for any _ ^ space Corollary 3.3 Every a-compact a-Hausdorff topological space is a-normal Lemma 3.5 For any fuzzifying a-topological space ðX; sÞ; Ta2 ðX; sÞ Ca ðX; sÞ ! Ta3 ðX; sÞ For the definition of Ta3 ðX; sÞ see 21, Definition 2.2] Proof Immediate, set A = {x} in the above lemma and sa =ABị ẳ h The above lemma is a generalization of the following corollary Locally a-compact spaces based on continuous valued logic 77 Corollary 3.4 Every a-compact a-Hausdorff topological space is a-regular Theorem 3.3 For any fuzzifying topological space (X, s) and A ˝ X, Ta2 ðX; sÞ Ca ðAÞ ! A Fa Proof For any {x} Ì Ac, we Ca({x}) = By Theorem 4.6 [1], have {x} \ A = / ! _  a à T2 ðX; sÞ ðCa ðAÞ ^ Ca ðfxgÞÞ and minðsa ðGÞ; sa ðHx ÞÞ : G\Hx ẳ/; A # G; x2Hx and La CA; s=Aị ẳ A max 0; Nax Gị ỵ Ca ðG; ðs=AÞ=GÞ À : ^_ x2AG # A Now, suppose that [(X, s) LaC A Fa] > k > Then for any x A, there exists B ˝ X such that X Nax ðBÞ þ Ca ðB; s=BÞ þ sa ðX À AÞ À > k: Set E = A \ B P(A) Then _ A X X Nax Eị ẳ Nax Cị P Nax Bị EẳC\B Assume [ bx ẳ fHx : A \ Hx ¼ /; x Hx g; and for any U P(E), we have _ _ sa =Aịa =EUị ẳ sa =ACị ẳ fxị Ac x2XnA UẳC\E and [ [ fxị \ A ẳ fxị \ Aị ẳ /: x2Ac So, Similarly, _ sa =Bịa =EUị ẳ x2Ac _ sa Dị ẳ UẳD\A\E fxị ẳ Ac _ sa Dị: UẳD\E sa Dị: UẳD\E Therefore _ ẵTa2 X; sị Ca Aị Thus, (sa/B)a/E = (sa/A)a/E and Ca(E, (s/A)/E) = Ca(E, (s/B)/ E) Furthermore, _ ẵE Fa =B ẳ sa =BB Eị ¼ sa =BðB \ Ec Þ ¼ sa ðDÞ sa Hx ị G\Hx ẳ/; A # G; x2Hx ^ _ sa Hx ị B\Ec ẳB\D x2Ac A\Hx ẳ/;x2Hx _ ¼ f2 Q x2A ^ x2A c bx _ f2 Q b x2Ac x ẳ Fa Aị: sa P sa X Aị ẳ Fa Aị: sa fxịị c [ x2Ac ! fxị _ ẳ f2 Q Since `(X, s) Ca A Fa fi (A, s/A) Ca (see [1], Theorem 4.1 (1)], from (*) we have for any x A that sa ðAc Þ b x2XnA x à The above theorem is a generalization of the following corollary Corollary 3.5 a-compact subspace of an a-Hausdorff topological space is a-closed Fuzzifying locally a-compactness Definition 4.1 Let X be a class of fuzzifying topological spaces A unary fuzzy predicate La C IðXÞ, called fuzzifying locally a-compactness, is given as follows: Since (X, s) LaC:ẳ("x)($B)((x Inta(B) Ca(B, s/B)) ẵx Inta Xị ¼ Nax ðXÞ ¼ 1, then LaC(X, s) P Ca(X, s) Therefore, `(X, s) Ca fi (X, s) LaC Also, since `(X, s) C fi (X, s) LC [24] and `(X, s) Ca fi (X, s) C[1], `(X, s) Ca fi (X, s) LC, which is a generalization of Corollary 4.4 [26] Theorem 4.1 For any fuzzifying topological space (X, s) and A ˝ X, `(X, s) LaC A Fa fi (A, s/A) LaC Proof We have La CX; sị ẳ sa Dị UẳC\ECẳD\A _ ẳ x2Ac S ðÃÞ ^_ x2XB # X X max 0; Nax Bị ỵ Ca B; s=Bị _ A max 0; Nax Gị ỵ Ca G; s=Aị=Gị G#A A P Nax Eị ỵ Ca E; s=Aị=Eị A ẳ Nax Eị ỵ Ca E; s=Bị=Eị X P Nax Bị ỵ ẵCa B; s=BÞ E Fa =B À X P Nax Bị ỵ Ca B; s=Bị ỵ ẵE Fa =B X P Nax Bị ỵ Ca B; s=Bị ỵ ẵA Fa > k: Therefore La CA; s=Aị ẳ ^_ A max 0; Nax Gị ỵ Ca G; s=Aị=Gị x2AG # A > k: Hence [(X, s) LaC A Fa] LaC(A, s/A) h As a crisp result of the above theorem we have the following corollary Corollary 4.1 Let A be an a-closed subset of locally a-compact space (X, s) Then A with the relative topology s/A is locally acompact The following theorem is a generalization of the statement ‘‘If X is an a-Hausdorff topological space and A is an a-dense a-locally compact subspace, then A is a-open’’, where A is an a-dense in a topological space X if and only if the a-closure of A is X 78 O.R Sayed Theorem 4.2 For any fuzzifying a-topological space (X, s) and A ˝ X, Hence Ta2 ðX; sÞ La CðA; s=AÞ ðCla ðAÞ XÞ ! A sa : Now, for any y Ac we have ^ X X ½Cla Aị X ẳ Nax Ac ị À Nay ðAc Þ: Proof Assume À Á sa Kx ị ỵ sa Bcx ỵ ẵCla Aị X > k: x2X ẵTa2 X; sị La CðA; s=AÞ ðCla ðAÞ XÞ > k > 0: Then  à La CðA; s=AÞ > k À Ta2 X; sị Cla Aị Xị ỵ ¼ k0 > k; i.e., ^_ X A max 0; Nax Bị ỵ Ca B; s=Aị=Bị > k0 : Nay ðAc Þ; x2AB # A Thus for any x A, there exists Bx ˝ A such that A Nax ðBx Þ i.e., _ Since (X, s) is a fuzzifying a-topological space, À Á À sa Kx ị ỵ sa Bcx sa ðKx Þ sa Bcx sa ðKx Þ ^ sa Bcx À Á Á XÀ sa Kx \ Bcx Nay Kx \ Bcx þ Ca ðBx ; ðs=AÞ=Bx Þ À > k : _ where À Á y Kx \ Bcx # Hx \ Hx \ Aịc ẳ Hx \ Hcx [ Ac ¼ Hx \ Ac # Ac : Therefore sa Kị ỵ Ca Bx ; s=Aị=Bx ị > k : H\A¼Bx x2K # H À Á < k < sa Kx ị ỵ sa Bcx ỵ ẵCla Aị X ẳ sa Kx ị ỵ sa Bcx ỵ ẵCla ðAÞ X À X such that Kx \ A = Hence there exists Kx Bx, sa(Kx) + Ca(Bx, (s/A)/Bx) À > k0 Therefore sa(Kx) > k0 (1) If for any x A thereS exists Kx such that x Kx ˝ Bx ˝ A, then K x ẳ A and sa Aị ẳ x2A S V Kx P sa ðK x Þ P k0 > k sa x2A x2A (2) If there exists x A such that À Á Kx \ Bcx /; sa Kx ị ỵ Ca Bx ; s=Aị=Bx Þ À > k0 : From the hypothesis  a à T2 ðX; sÞ La CðA; s=AÞ ðCla ðAÞ XÞ > k > 0;  à we have Ta2 ðX; sÞ ðCla ðAÞ XÞ So sa Kx ị ỵ Ca Bx ; s=Aị=Bx ị ỵ ẵTa2 X; sị Cla Aị a contradiction So, case (2) does not hold We complete the proof h Theorem 4.3 For any fuzzifying a-topological space (X, s), X Ta2 ðX;sÞðLa CðX;sÞÞ2 ! 8x8U U Nax X ! 9V V Nax ^Cla Vị#U^Ca Vị ; where (LaC(X, s))2:ẳLaC(X, s) LaC(X, s) Proof We need to show that for any x and U, x U, X Ta2 ðX; sÞ ðLa CðX; sÞÞ2 Nax ðUÞ XÞ > k: X Nay Ac ị ỵ Nay Ac ị ẳ 0; _ X Nax ðVÞ X Nax ðVc Þ ! ^ Ca ðV; s=VÞ : y2Uc V#X Therefore ^ ^ X sa Kx ị ỵ Ca Bx ; s=Aị=Bx ị ỵ Ta2 X; sị ỵ ẵCla Aị Xị À > k: X ðsa =AÞa =Bx ðUÞ ¼ U¼C\Bx ¼ _ ¼ ðÃÞ Since (X, s) is fuzzifying a-topological space, sa =ACị _ sa Dị UẳC\Bx CẳD\A _ X Ta2 X; sị ỵ Nax Cị ỵ La CX; sịị2 ỵ Nax Uị > k: Since _ Assume that Ta2 ðX; sÞ ðLa CðX; sÞÞ2 Nax ðUÞ > k > Then for any x X there exists C such that sa ðDÞ ¼ sa =Bx ðUÞ; U¼D\Bx Ca ðBx ; ðs=AÞ=Bx Þ ¼ Ca ðBx ; s=Bx Þ: From Theorem 3.3 we have À Á sa Bcx P Ta2 ðX; sÞ Ca Bx ; s=Bx ị P Ta2 X; sị ỵ Ca ðBx ; s=Bx Þ À 1: X X X X X X Nax Cị ỵ Nax Uị Nax ðCÞ Nax ðUÞ Nax ðCÞ ^ Nax Uị _ X Nax C \ Uị ẳ sa ðWÞ: x2W # C\U Therefore there exists W such that x W ˝ C \ U, and Ta2 X; sị ỵ La CX; sịị2 ỵ sa Wị > k By Lemmas 3.3 and 3.5 we have Ta2 ðX; sÞ Ta2 ðC; s=CÞ and Ta2 ðC; s=Cị ỵ Ca C; s=Cị Ta2 C; s=CÞ Ca ðC; s=CÞ Ta3 ðC; s=CÞ: Thus Ta3 X; sị ỵ Ca C; s=Cị ỵ sa Wị À > k Since for any x W ˝ U, we have Locally a-compact spaces based on continuous valued logic Ta3 ðC; s=CÞ À sa =CðWÞ þ _ ^ C Nax ðGÞ ^ G#C Theorem 4.4 For any fuzzifying a-topological space (X, s), Ta2 ðX; sÞ ðLa CðX; sÞÞ2 ! Ta3 ðX; sÞ !! C Nay ðC À GÞ 79 ; Proof By Theorem 4.3, for any x U, we have y2CÀW so there exists G, x G ˝ W such that !! ^ aC aC Nx ðGÞ ^ Ny ðC À GÞ _ X Nax ðVÞ ^ h i X P Ta2 ðX; sÞ ðCa ðC; s=CÞÞ2 Nax ðUÞ : P Ta3 C; s=Cị ỵ sa =CWị P Ta3 C; s=Cị ỵ sa Wị Thus and !! ^ aC Nx ðGÞ ^ aC X Nax Uị ỵ ỵ Ca C; s=Cị > k: Ny ðC À GÞ _ C X X Nax Dị ẳ Nax G [ Cc ị > k0 Furthermore, for any _ X X Nay ðGc [ Cc Þ ¼ Nay ðGc Þ > k0 D\C¼C\Gc ¼ X Nax ððG X Nax ðG c c X Nax ðCÞ [ C Þ \ CÞ P [C Þ^ >k: W X Since Nay Gc ị ẳ sa Bc ị > k0 , for any y C À W, there x2Bc # Gc exists Bcy such that y Bcy # Gc and sa ðBcy Þ > k0 Set S Bc ¼ Bcy Then C À W ˝ Bc ˝ Gc and sa ðBc Þ P V y2CÀWc sa ðBy Þ P k0 Again, set V = B \ C, then y2CÀW V ˝ (C À W)c \ C = (Cc [ W) \ C = C \ W = W ˝ U \ C and Vc = Bc [ Cc.Since (X, s) is fuzzifying a-topological space, X X X X Nax Vị ẳ Nax B \ CÞ P Nax ðBÞ ^ Nax ðCÞ X X P Nax ðGÞ ^ Nax ðCÞ > k: ð1Þ h X X where U NaA :ẳ 8xị x A ^ U Nax h i X Ta3 ðX; sÞ La CðX; sÞ Ca ðA; s=AÞ NaA ðUÞ _ X NaA ðVÞ ^ sa ðVc Þ ^ Ca ðV; s=VÞ : V#U Indeed, if h i X Ta3 ðX; sÞ La CðX; sÞ Ca ðA; s=AÞ NaA ðUÞ > k > 0; then for any x A, there exists C P(X) such that h i X X Ta3 ðX; sÞ Nax ðCÞ Ca ðC; s=CÞ Ca ðA; s=AÞ NaA ðUÞ > k: Since (X, s) is fuzzifying a-topological space, _ By ðÃÞ and Theorem3:3; sa ðCc Þ P Ta2 ðX; sÞ Ca C; s=Cị P Ta2 X; sị ỵ Ca C; s=CÞ À P k0 : So sa ðVc Þ X X X Therefore ð2Þ Finally,  a à T3 X; sị ỵ sa Wị > k ỵ Ca C; s=Cị Ca A; s=Aị ẳ k0 P k: P X Nay Vc ị c ẳ sa ðV Þ P k ð3Þ y2Vc Thus by (1)–(3), for any x U, there exists V ˝ U such that V aX c X Ny ðV Þ P k and Ca(V, s/V) P k So Nax ðVÞ > k; y2Uc ! V aX c W X Nax ðVÞ ^ Ny ðV Þ ^ Ca ðV; s=VÞ P k h y2Uc X Then there exists W such that x W ˝ C \ U, and  a à T3 ðX; sÞ sa ðWÞ Ca ðC; s=CÞ Ca ðA; s=AÞ > k: P sa =CðC À Vị ỵ Ca C; s=Cị ^ X P Nax ðCÞ ^ NaA ðUÞ P Nax ðCÞ NaA Uị: P kand Ca V; s=Vị ẳ Ca V; s=Cị=Vị X Nay ðVc Þ X x2W # C\U P k0 ; i:e:; sa Vc ị ỵ Ca C; s=Cị P sa Vc ị ỵ Ca C; s=Cị P k X sa Wị ẳ Nax C \ Uị P Nax Cị ^ Nax Uị ẳ sa Bc [ Cc Þ P sa ðBc Þ ^ sa ðCc Þ V#X y2Uc Proof We only need to show that for any A, U P(X), and X Nax ðGÞ ! h i X Nay ðVc Þ P Ta2 ðX; sÞ ðCa ðC; s=CÞÞ2 ; X Ta3 ðX; sÞ La CðX; sÞ ! 8A8U U NaA Ca ðA; s=AÞ X ! 9V V # U ^ U NaA ^ sa ðVc ị ^ Ca V; s=Vị ; ẳ k ỵ À Ca ðC; s=CÞ P k: C ^ Theorem 4.5 For any fuzzifying a-topological space (X, s), D\C¼G y C W; Nay C Gị ẳ X Nax ðVÞ ^ h i  à i.e., Ta3 ðX; sÞ P Ta2 ðX; sÞ ðCa ðC; s=CÞÞ2 Thus Nax Gị ẳ _ x2V # U y2CW y2Uc X Nay ðVc Þ y2Uc x2V # U y2CÀW ^ ^ ðÃÞ Since for any  à x W; Ta3 X; sị sa Wị ỵ _ B#W we have X Nax ðBÞ ^ ^ y2Wc ! X Nay ðBc Þ ; 80 O.R Sayed _ ^ X Nax ðBÞ ^ ! X > k0 : Nay ðBc Þ y2Wc B#W Thus there exists Bx such that x Bx ˝ W ˝ C \ U and for any y Wc, we have Á XÀ X Nay Bcx > k0 ; Nax ðBx Þ > k0 : Since Á XÀ Nay Bcx ¼ _ x2Gc c Gcxy # Bcx then for any y W , there exists Gxy such that x S c and sa Gcxy > k0 Set Gcx ¼ Gxy , then Wc # Gcxy # Bcx and y2Wc À Á V c X sa Gxy P k0 Since Gx Bx ; Nax ðGx Þ P sa Gcx P W X Nax ðBx Þ > k0 , i.e., sa ðHÞ > k0 Thus there exists Hx such x2H # Gx that x Hx ˝ Gx and sa(Hx) > k0 Hence for any x A, there exists S Hx and G Sx such that x Hx ˝ Gx ˝ U, sa(Hx) > k and W Gx Hx A We dene R IPAịị as follows: x2A RDị ẳ Let W x2A _ < sa ðHx Þ; there exists Hx such that Hx \ A ¼ D; : }6R ẵKR; Aị ẳ ^_ RBị ẳ x2Ax2B ^_ RDị ẳ ^_ _ sa Hx0 ị P k0 x2Ax2DHx0 \AẳD x2Ax2D Sn V c sa V ị P iẳ1 Gxi , sa Gcx0 i 16i6n Vc ¼ and Tn c i¼1 Gx0i ; A # V # U, Since P k > k for and any Sn x A, Gx ˝ W ˝ C \ U ˝ C, we have V ¼ i¼1 Gx0i # W # C W sa ðDÞ P sa ðVc Þ P k0 Thus Because sa =CC Vị ẳ D\CẳC\Vc by (*), sa/C(C À V) + Ca(C, s/C) À > k, and by Theorem 4.1 [1], Ca(V, s/V) = Ca(V, s/C/V) P [Ca(C, s/C) sa/ C(C À V)] > k Finally, we have for any x A, X Nax Vị ẳ n [ Gx0i X Nax ^ ! i¼1 P X Nax n [ ! Hx0i P sa i¼1 n [ Hx0i ! i¼1 sa Hx0i P k > k: 16i6n and V aX X X Nx ðVÞ P k Therefore NaA ðVÞ ^ sa Vc ị^ So, NaA Vị ẳ Ca V; s=Vị P k.x2A Thus _ X NaA ðVÞ ^ sa ðVc Þ ^ Ca ðV; s=VÞ P k: à V#U ẵR # sa =A ẳ ^ min1; RBị ỵ sa =ABịị B#X ẳ ^ 1; _ sa Hx ị ỵ Hx \AẳB B#X ! _ sa Hị ẳ 1: ẵK}; Aị FF}ị > l ỵ Ka R; Aị P l ỵ k0 x2Ax2E ^ ^_ fd : Fð}d Þg À > k À ; and }Eị ^ > k ỵ fd : Fð}d Þg: Proof If [LaC(X, s) Ca(f) O(f)] > k > 0, then for any X x X, there exists U X, such that ẵNax Uị W X sa ðVÞ, Ca ðU; s=UÞ Ca ðfÞ Ofị > k Since Nax Uị ẳ x2V # U P k : }Eị ỵ Theorem 4.6 Let (X, s) and (Y, r) be two fuzzifying topological spaces and f YX be surjective Then ` LaC(X, s) Ca(f) O(f) fi LC(Y, r) For the definition of O(f), see [17] H\A¼B So, Ka ðR; Aị ẳ ẵKR; Aị P k0 By (*), Thus ^_ Vẳ P Ka R; Aịỵ Then nite i¼1 otherwise: Ca(A, s/A) = l > l À e(e > 0) ẵK}; Aị FF}ị > l , where exists n n [ [ Hx0i # Gx0i : Hx \A¼D 0; there and Set # Bcx so i¼1 iẳ1 sa Gc ị > k0 ; y2Wc i.e., Hx0 \ A }b By F(}b), Hx01 ; Hx02 ; ; Hx0n such that n [ Hx0i A x2Ax2E so there exists V0 ˝ X such that x V0 ˝ U and [sa(V0 ) Ca(U, s/U) Ca(f) O(f)] > k By Theorem 4.3 in [1], [Ca(U, s/U) Ca(f)] [C(f(U), r/f(U))] and ẵsV0 ị Ofị ẳ max0; sV0 ị ỵ Ofị 1ị ẳ max0; sV0 ị ^ ỵ min1; sV0 ị ỵ rfVịịị 1ị V#X Hence there exists b > such that F(}b) and ^_ }Dị > k ỵ b: max0; sV0 ị ỵ sV0 ị ỵ rfVịị 1ị ẳ rfVịị NYfxị fV0 ịị NYfxị fUịị: x2Ax2D Therefore for any x A, there exists Dx ˝ A such that }(Dx) > k À e + b and [ Dx # A: Since f is surjective, LCY; rị ẳ LCfXị; rị ^ _ ẳ x2A h i NYy U0 ị ẵCU0 ; r=U0 ị y2fxị # fXịU0 ẳfUị # fXị Suitably choose e such that k À e > 0, then }(Dx) > b > Since P RðDx Þ P }ðDx Þ > 0; Dx ẳ Hx0 \ A; P h ^ NYfxị fUịị ẵCfUị; r=fUịị i y2fxị # fXị ^ ẵsV0 ị OðfÞ Ca ðU; s=UÞ Ca ðfÞ P k: à y2fðxÞ # fðXÞ Locally a-compact spaces based on continuous valued logic 81 Theorem 4.7 Let (X, s) and (Y, r) be two fuzzifying topological spaces and f YX be surjective Then `LaC(X, s) Ia(f) Oa(f) fi LaC(Y, r) Xk is locally a-compact and all but finitely many Xk are a-compact Proof By Theorem 4.3 in [1], the proof is similar to the proof of Theorem 4.6 h Conclusion Theorems 4.6 and 4.7 are a generalization of the following corollary Corollary 4.2 Let (X, s) and (Y, r) be two topological spaces and f:(X, s) fi (Y, r) be surjective mapping If f is an acontinuous (resp a-irresolute), open (resp a-open) and X is locally a-compact, then Y is locally compact (resp locally acompact) space Theorem 4.8 Let {(Xs, ss):s S} be a family of fuzzifying topological spaces, then Y Y Xs ; ðsa Þs s2S La C ! ! 8sðs s2S S ^ La CðXs ; ðsa Þs Þ ^ 9TðTbS ^ 8tðt S À T ^ Ca ðXt ; st ÞÞÞ: The present paper investigates topological notions when these are planted into the framework of Ying’s fuzzifying topological spaces (in semantic method of continuous valued-logic) It continue various investigations into fuzzy topology in a legitimate way and extend some fundamental results in general topology to fuzzifying topology An important virtue of our approach (in which we follow Ying) is that we define topological notions as fuzzy predicates (by formulae of Łukasiewicz fuzzy logic) and prove the validity of fuzzy implications (or equivalences) Unlike the (more wide-spread) style of defining notions in fuzzy mathematics as crisp predicates of fuzzy sets, fuzzy predicates of fuzzy sets provide a more genuine fuzzification; furthermore the theorems in the form of valid fuzzy implications are more general than the corresponding theorems on crisp predicates of fuzzy sets The main contributions of the present paper are to give characterizations of fuzzifying a-compactness Also, we define the concept of locally a-compactness of fuzzifying topological spaces and obtain some basic properties of such spaces There are some problems for further study: Proof It suffices to show that " # Y ^ Y _ ^ La C Xs ; s2S ðsa Þs La CðXs ; ðsa Þs Þ ^ Ca ðXt ; st Þ : s2S s2S TbSt2SÀT From Theorem 4.7 and Lemma 3.1 we have for any t S, Y h Y Y Y La C X ðs X ðs s; a Þs ¼ La C s; a Þs s2S s2S s2S s2S Ca ðpt Þ Oa ðpt Þ La CðXt ; st Þ: So, Y ^ Y La CðXt ; st Þ P La C Xs ; ðsa Þs : s2S s2S Acknowledgement t2S By Theorem 3.2 we have _ ^ TbSt2SÀT Y Ca U; s2S ðsa Þs =U _ Ca ðXt ; st Þ P U# Q _ P U# s2S Q s2S Xs Xs X # ^ P X# Q ¼ La C s2S s2S Xs U # s2S Q Xs ; s2S Q s2S X Nax ðUÞÞ5 h Y i X Ca U; s2S ðsa Þs =U Nax ðUÞ Xs Y s2S ðsa Þs : Therefore La C Y Xs ; s2S " # ^ _ ^ ðs Þ L CðX ; s Þ ^ C ðX ; s Þ : à a s a t t a t t s2S Y t2S Author would like to express his sincere thanks to the referees for giving valuable comments which helped to improve the presentation of this paper Xs Xs _ Y _ X# Q h Y i X Ca U; s2S ðsa Þs =U Nax ðUÞ _ (1) One obvious problem is: our results are derived in the Łukasiewicz continuous logic It is possible to generalize them to more general logic setting, like residuated lattice-valued logic considered in [27,28] (2) What is the justification for fuzzifying locally a-compactness in the setting of (2, L) topologies (3) Obviously, fuzzifying topological spaces in [14] form a fuzzy category Perhaps, this will become a motivation for further study of the fuzzy category (4) What is the justification for fuzzifying locally a-compactness in (M, L)-topologies etc TbSt2SÀT We can obtain the following corollary in crisp setting Corollary 4.3 Let {X Qk:k K} be a family of nonempty topological spaces If k2K Xk is locally a-compact, then each References [1] O.R Sayed, a-Iresoluteness and a-compactness based on continuous valued logic, J Eg Math Soc 20 (2012) 116–125 [2] M.S Ying, A new approach for fuzzy topology (I), Fuzzy Sets and Systems 39 (1991) 303321 [3] U Hoăhle, Many Valued Topology and its Applications, Kluwer Academic Publishers, Dordrecht, 2001 [4] U Hoăhle, S.E Rodabaugh, in: Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Handbook of Fuzzy Sets Series, vol 3, Kluwer Academic Publishers, Dordrecht, 1999 [5] U Hoăhle, S.E Rodabaugh, A Sostak, Special issue on fuzzy topology, Fuzzy Sets Syst 73 (1995) 1–183 [6] T Kubiak, On Fuzzy Topologies, Ph.D Thesis, Adam Mickiewicz University, Poznan, Poland, 1985 [7] Y.M Liu, M.K Luo, Fuzzy Topology, World Scientific, Singapore, 1998 82 [8] G.J Wang, Theory of L-Fuzzy Topological Spaces, Shanxi Normal University Press, Xian, 1988 (in Chinese) [9] C.L Chang, Fuzzy topological spaces, J Math Anal Appl 24 (1968) 182–190 [10] J.A Goguen, The fuzzy Tychonoff theorem, J Math Anal Appl 43 (1973) 182–190 [11] J.L Kelley, General Topology, Van Nostrand, New York, 1955 [12] U Hoăhle, Upper semicontinuous fuzzy sets and applications, J Math Anal Appl 78 (1980) 659673 [13] U Hoăhle, A Sostak, Axiomatic foundations of xed-basis fuzzy topology, in: U Hoăhle, S.E Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Handbook of Fuzzy Sets Series, vol 3, Kluwer Academic Publishers, Dordrecht, 1999, pp 123–272 [14] S.E Rodabaugh, Categorical foundations of variable-basis fuzzy topology, in: U Hoăhle, S.E Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Handbook of Fuzzy Sets Series, vol 3, Kluwer Academic Publishers, Dordrecht, 1999, pp 273–388 [15] J.B Rosser, A.R Turquette, Many-Valued Logics, NorthHolland, Amsterdam, 1952 [16] M.S Ying, A new approach for fuzzy topology (II), Fuzzy Sets Syst 47 (1992) 221–223 [17] M.S Ying, A new approach for fuzzy topology (III), Fuzzy Sets Syst 55 (1993) 193–207 O.R Sayed [18] F.H Khedr, F.M Zeyada, O.R Sayed, a-Continuity and cacontinuity in fuzzifying topology, Fuzzy Sets Syst 116 (2000) 325–337 [19] D Qiu, Fuzzifying topological linear spaces, Fuzzy Sets Syst 147 (2004) 249–272 [20] D Qiu, Characterizations of fuzzy finite automata, Fuzzy Sets Syst 141 (2004) 391–414 [21] O.R Sayed, a-Separation axioms based on Łukasiewicz logic, Hacettepe J Math Stat., accepted for publication [22] M.S Ying, Compactness in fuzzifying topology, Fuzzy Sets Syst 55 (1993) 79–92 [23] O.R Sayed, On Fuzzifying Topological Spaces, Ph.D Thesis, Assiut University, Egypt, 2002 [24] J Shen, Locally compactness in fuzzifying topology, J Fuzzy Math (4) (1994) 695–711 [25] S.N Maheshwari, S.S Thakur, On a-compact spaces, Bull Inst Math Acad Sinica 13 (4) (1985) 341–347 [26] T Noiri, G Di Maio, Properties of a-compact spaces, III Convegno Topologia, Trieste, Giugno 1986, Supp Rend Circ Mater Palermo, Seric II 18 (1988) 359–369 [27] M.S Ying, Fuzzifying topology based on complete residuated lattice-valued logic (I), Fuzzy Sets Syst 56 (1993) 337–373 [28] M.S Ying, Fuzzy topology based on residuated lattice-valued logic, Acta Math Sinica 17 (2001) 89–102 ... K} be a family of nonempty topological spaces If k2K Xk is locally a -compact, then each References [1] O.R Sayed, a-Iresoluteness and a-compactness based on continuous valued logic, J Eg Math Soc... Definition 2.2] Proof Immediate, set A = {x} in the above lemma and sa =AðBÞ ¼ h The above lemma is a generalization of the following corollary Locally a -compact spaces based on continuous valued logic. .. s=UÞ Ca ðfÞ P k: à y2fðxÞ # fðXÞ Locally a -compact spaces based on continuous valued logic 81 Theorem 4.7 Let (X, s) and (Y, r) be two fuzzifying topological spaces and f YX be surjective Then