- Email: [email protected]
More on θ-compact fuzzy topological spaces
Chaos, Solitons and Fractals 27 (2006) 1157–1161 www.elsevier.com/locate/chaos
More on h-compact fuzzy topological spaces Erdal Ekici Department of Mathematics, Canakkale Onsekiz Mart University, Terzioglu Campus, 17020 Canakkale, Turkey Accepted 8 April 2005
Abstract Recently, El-Naschie has shown that the notion of fuzzy topology may be relevant to quantum particle physics in connection with string theory and e1 theory. In 2005, Caldas and Jafari have introduced h-compact fuzzy topological spaces. The purpose of this paper is to investigate further properties of h-compact fuzzy topological spaces. Moreover, the notion of h-closed fuzzy topological spaces is introduced and properties of it are obtained. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction In the literature there are many papers on compactness. Compactness occupies a very important place in fuzzy topology and so do some of its forms [3,6,9,11], etc. Moreover, El-Naschie [4,5] has shown that the notion of fuzzy topology may be relevant to quantum particle physics in connection with string theory and e1 theory. Also, applicability of the new concepts in this paper to superstrings and e1 space time could be probably possible in the near future. In [7,8], Mukherjee and Sinha introduced the concept of h-open sets in fuzzy setting. In [1], the notion of h-compactness is introduced by Caldas and Jafari. The aim of this paper is devoted to introduce the further properties of h-compact fuzzy topological space and to introduce h-closed fuzzy topological space. The notion of h-closedness comes out to be strictly weaker than fuzzy almost compactness. Also, the relationships between fuzzy h-compactness and fuzzy filterbases and the relationships between fuzzy filterbases and fuzzy h-closedness are obtained.
2. Preliminaries Throughout this paper X and Y mean fuzzy topological spaces. A fuzzy point xp in X is a fuzzy set having support x 2 X and value p 2 (0, 1] [10]. The complement and the support of a fuzzy set c denoted by cc and S(c), respectively. For two fuzzy sets c and q, we shall write cqqðcqqÞ to mean that c is quasi coincident (not quasi coincident) with q, i.e, there exists x 2 X such that c(x) + q(x) > 1 (c(x) + q(x) 6 1) [10]. Definition 1 ([7,8]). A fuzzy point xp in a fuzzy topological space X is said to be a fuzzy h-cluster point of a fuzzy set k if and only if for every fuzzy open q-neighbourhood l of xp, cl(l) is q-coincident with k. The set of all fuzzy h-cluster E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.077
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points of k is called the fuzzy h-closure of k and is denoted by h-cl(k). A fuzzy set k is fuzzy h-closed if and only if k = h-cl(k). The complement of a fuzzy h-closed set is called fuzzy h-open. The h-interior of k denoted by h-int(k) is defined as h-intðkÞ ¼ fxp : for some fuzzy open q-neighbourhood b of xp ; clðbÞ 6 kg. Definition 2. A fuzzy topological space X is said to be almost compact [3] if and only if every open cover of X has a finite subcollection whose closures cover X. Lemma 3. Let q be a fuzzy set in a fuzzy topological space X, then (1) (2) (3) (4) (5)
q is a fuzzy h-open if and only if q = h-int(q), Xnh-int(q) = h-cl(Xnq), h-int(Xnq) = Xnh-cl(q), h-cl(q) is a fuzzy closed set but not necessarily a fuzzy h-closed set, h-int(q) is a fuzzy open set but not necessarily a fuzzy h-open set.
Remark 4. Let q be a fuzzy set in a fuzzy topological space X, then (1) (2) (3) (4)
cl(q) 6 h-cl(q), h-int(q) 6 int(q), if q is fuzzy open set, then cl(q) = h-cl(q), if q is fuzzy closed set, then int(q) = h-int(q).
Definition 5 [2]. A fuzzy topological space (X, s) is called an extremally disconnected space if cl(g) 2 s for every fuzzy open set g.
3. Properties of fuzzy h-compact space In this section, further properties of fuzzy h-compact spaces are investigated. Definition 6. A collection W of fuzzy sets in a fuzzy topological space X is said to be cover of a fuzzy set c of X if and only if (_g2W)(x) = 1X, for every x 2 S(c). A fuzzy cover W of a fuzzy set c in a fuzzy topological space X is said to have a finite subcover if and only if there exists a finite subcollection D = {g1, . . ., gn} of W such that ð_nj¼1 gj ÞðxÞ P cðxÞ, for every x 2 S(c). Definition 7 [1]. A fuzzy topological space X is said to be fuzzy h-compact if and only if for every family W of h-open fuzzy sets such that _g2Wg = 1X there exists a finite subfamily D W such that _g2Dg = 1X. Definition 8 [1]. A fuzzy set c in a fuzzy topological space X is said to be fuzzy h-compact relative to X if and only if for every family W of h-open fuzzy sets such that _g2Wg P c(x) there exists a finite subfamily D W such that _g2Dg P c(x) for every x 2 S(c). Definition 9. A collection of fuzzy subsets n of a fuzzy topological space X is said to form a fuzzy filterbases if and only if for every finite collection {gj : j = 1, . . ., n}, ^nj¼1 gj 6¼ 0X . Theorem 10. A fuzzy topological space X is fuzzy h-compact if for every filterbases n in X, ^l2nh-cl(l) 5 0X. Proof. Let W be a h-open fuzzy set cover of X and W has no a finite subcover. Then for every finite subcollection {g1, . . ., gn} of W, there exists x 2 X such that gj(x) < 1 for every j = 1, . . ., n. Then gcj ðxÞ > 0, so that ^nj¼1 gcj ðxÞ 6¼ 0X . Thus, fgcj ðxÞ : gj 2 Wg forms a filterbases in X. Since W is h-open fuzzy set cover of X, then ð_gj 2W gj ÞðxÞ ¼ 1X for every x 2 X and hence ^gj 2W h-clðgci ðxÞÞ ¼ ^gj 2W gcj ðxÞ ¼ 0X ;
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which is a contradiction. Then every h-open fuzzy set cover of X has a finite subcover and hence X is fuzzy hcompact. h Theorem 11. If a fuzzy extremally disconnected topological space X is fuzzy h-compact, then for every filterbases n in X, ^l2nh-cl(l) 5 0X. Proof. Suppose there exists of filterbases n such that ^l2nh-cl(l) = 0X, so that (_l2n(h-cl(l))c)(x) = 1X for every x 2 X and hence W ¼ fðh-cllÞc : l 2 ng is a h-open fuzzy set cover of X. Since X is fuzzy h-compact, then W has a finite subcover. Then ð_nj¼1 ðh-clðlj ÞÞc ÞðxÞ ¼ 1X and hence ð_nj¼1 lcj ÞðxÞ ¼ 1X , so that ^nj¼1 lj ¼ 0X which is contradiction, since the lj are members of filterbases n. Therefore ^l2nh-cl(l) 5 0X for every filterbases n. h Corollary 12. Let X be a fuzzy extremally disconnected topological space. Then X is fuzzy h-compact if and only if for every filterbases n in X, ^l2nh-cl(l) 5 0X. Theorem 13. A fuzzy set c in a fuzzy topological space X is fuzzy h-compact relative to X if for every filterbases n such that every finite of members of n is quasi coincident with c, (^l2nh-cl(l)) ^ c 5 0X. Proof. Let c not be fuzzy h-compact relative to X, then there exists a h-open fuzzy set W cover of c such that W has no finite subcover D. Then ð_gj 2D gj ÞðxÞ < cðxÞ for some x 2 S(c), so that ð^nj 2D gcj ÞðxÞ > cc ðxÞ P 0 and hence n ¼ fgcj : gj 2 Wg forms a filterbases and ^gj 2D gcj qc. By hypothesis ð^gj 2D h-clðgcj ÞÞ ^ c 6¼ 0X and hence ð^gj 2D gcj Þ ^ c 6¼ 0X . Then for some x 2 S(c), ð^gj 2W gcj ÞðxÞ > 0X , that is ð_gj 2W gj ÞðxÞ < 1X , which is a contradiction. Hence c is fuzzy h-compact relative to X. h Theorem 14. If a fuzzy set c in a fuzzy extremally disconnected topological space X is fuzzy h-compact relative to X, then for every filterbases n such that every finite of members of n is quasi coincident with c, (^l2nh-cl(l)) ^ c 5 0X. Proof. Suppose that there exists a filterbases n such that every finite of members of n is quasi coincident with c and (^l2nh-cl(l)) ^ c = 0X. Then for every x 2 S(c), (^l2nh-cl(l))(x) = 0X and hence (_l2n(h-cl(l))c)(x) = 1X for every x 2 S(c). Thus W = {(h-cll)c : l 2 n} is h-open fuzzy set cover of c. Since c is fuzzy h-compact relative to X, then there exists a finite subcover, say fðh-clðl1 ÞÞc ; . . . ; ðh-clðln ÞÞc g; such that ð_nj¼1 ðh-clðlj ÞÞc ÞðxÞ P cðxÞ for every x 2 S(c). Hence ð^nj¼1 h-clðlj ÞÞðxÞ 6 cc ðxÞ for every x 2 S(c), so that ^nj¼1 ðh-clðlj ÞÞqc, which is a contradiction. Therefore, for every filterbases n such that every finite of members of n is quasi coincident with c, (^l2nh-cl(l)) ^ c 5 0X. h Corollary 15. Let X be a fuzzy extremally disconnected topological space. Then a fuzzy set c in X is fuzzy h-compact relative to X if and only if for every filterbases n such that every finite of members of n is quasi coincident with c, (^l2nhcl(l)) ^ c 5 0X. Definition 16. A function f : X ! Y is said to be fuzzy h-open if and only if the image of every h-open fuzzy set in X is hopen in Y. Theorem 17. Let f : X ! Y be a fuzzy h-open bijective function and Y is fuzzy h-compact, then X is fuzzy h-compact. Proof. Let {gj : j 2 J} be a collection of h-open fuzzy set cover of X, then {f(gj) : j 2 J} is h-open fuzzy set covering of Y. Since Y is fuzzy h-compact, there exists a finite subset F J such that {f(gj) : j 2 F} is a cover of Y. We have 1X ¼ f 1 ð1Y Þ ¼ f 1 ðf ð_j2F gj ÞÞ ¼ _j2F gj and therefore, X is fuzzy h-compact.
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4. A new form of fuzzy compactness and properties of this form In this section, the concept of fuzzy h-closed topological spaces is introduced and properties of this notion are investigated. Definition 18. A fuzzy set c in a fuzzy topological space X is said to be a h-q-neighbourhood of a fuzzy point xp in X if there exists a h-open fuzzy set g 6 c such that xpqg. Theorem 19. Let xp be a fuzzy point in a fuzzy topological space X and c be any fuzzy set of X, then xp 2 h-cl(c) if and only if for every h-q-neighbourhood q of xp, qqc. qc. Then there exists a h-open fuzzy set Proof. ()): Let xp 2 h-cl(c) and there exists a h-q-neighbourhood q of xp, q g 6 q in X such that xpqg, which implies gqc and hence c 6 gc. Then h-cl(c) 6 gc. Since xp 62 gc, then xp 62 h-cl(c), which is a contradiction. ((): Obvious. h Definition 20. A fuzzy topological space X is said to be h-closed if and only if for every family W of h-open fuzzy set such that _g2Wg = 1X there exists a finite subfamily D W such that (_g2Dh-cl(g))(x) = 1X, for every x 2 X. Remark 21. The following implication holds for a fuzzy topological space X: fuzzy almost compact ) fuzzy h-closed. Question: Is there an example showing that fuzzy h-closedness does not imply fuzzy almost compactness? Theorem 22. A fuzzy extremally disconnected topological space X is h-closed if and only if for every fuzzy h-open filterbases n in X, ^l2nh-cl(l) 5 0X. Proof. Let W be a h-open fuzzy set cover of X and let for every finite subfamily D of W, (_g2Dh-cl(g))(x) < 1X for some x 2 X. Then (^g2Dh-cl(gc))(x) > 0X for some x 2 X. Thus, fðh-clðgÞÞc : g 2 Wg ¼ n forms a fuzzy h-open filterbases in X. Since W is a h-open fuzzy set cover of X, then ^g2Wgc = 0X which implies ^g2Whcl(h-cl(g)c) = 0X, which is a contradiction. Then every h-open fuzzy set W cover of X has a finite subfamily D such that (_g2Dh-cl(g))(x) = 1X for every x 2 X. Hence, X is h-closed. Conversely, suppose there exists a fuzzy h-open filterbases n in X such that ^l2nh-cl(l) = 0X, so that (_l2n(hcl(l))c)(x) = 1X for every x 2 X and hence W = {(h-cll)c : l 2 n} is a h-open fuzzy set cover of X. Since X is h-closed, then W has a finite subfamily D such that ð_l2D h-clððh-clðlÞÞc ÞÞðxÞ ¼ 1X for every x 2 X, and hence ^l2D(h-cl((h-cl(l))c))c = 0X. Thus, ^l2Dl = 0X which is a contradiction, since all the l are members of filterbases. h Definition 23. A fuzzy set c in a fuzzy topological space X is said to be h-closed relative to X if and only if for every family W of h-open fuzzy sets such that _g2Wg P c, there exists a finite subfamily D W such that (_g2Dhcl(g))(x) P c(x) for every x 2 S(c). Theorem 24. A fuzzy subset c in a fuzzy extremally disconnected topological space X is h-closed relative to X if and only if every fuzzy h-open filterbases n in X, (^l2nh-cl(l)) ^ c = 0X, there exists a finite subfamily k of n such that ð^l2k lÞ qc. Proof. Let c be a h-closed relative to X, suppose n is a fuzzy h-open filterbases in X such that for every finite subfamily k of n, (^l2kl)qc, but (^l2nh-cl(l)) ^ c = 0X. Then for every x 2 S(c), (^l2nh-cl(l))(x) = 0X and hence (_l2n(hcl(l))c)(x) = 1X for every x 2 S(c). Then W = {(h-cll)c : l 2 n} is a h-open fuzzy set cover of c and hence there exists a finite subfamily k n such that (_l2kh-cl(h-cl(l))c) P c, so that ^l2k ðh-clðh-clðlÞÞc Þc ¼ ^l2k h-intðh-clðlÞÞ 6 cc and hence ^l2kl 6 cc. Then ^l2k lqc which is a contradiction.
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Conversely, let c not be a h-closed fuzzy set relative to X, then there exists a h-open fuzzy set W cover of c such that every finite subfamily D W, ð_l2D h-clðgÞÞðxÞ 6 cðxÞ for some x 2 S(c) and hence (^l2D(h-cl(g))c)(x) > cc(x) P 0 for some x 2 S(c). Thus n = {(h-clg)c : g 2 W} forms a fuzzy h-open filterbases in X. Let there exists a finite subfamily {(hcl(g))c : g 2 D} such that ð^g2D ðh-clðgÞÞc Þqc. Then c 6 _g2Dh-cl(g). So there exists a finite subfamily D W such that _g2Dh-cl(g) P c which is a contradiction. Then for each finite subfamily k = {(h-clg)c : g 2 D} of n, We have (^g2D(hclg)c)qc. Hence by the given condition (^g2Wh-cl((h-cl(g))c)) ^ c 5 0X, so there exists x 2 S(c) such that (^g2Wh-cl((hcl(g))c))(x) > 0X. Then ð_g2W ðh-clððh-clðgÞÞc ÞÞc ÞðxÞ ¼ ð_g2W h-intðh-clðgÞÞÞðxÞ < 1X and hence, (_g2Wg)(x) < 1X which contradicts the fact that W is a h-open fuzzy set cover of c. Therefore, c is fuzzy h-closed relative to X. h Definition 25. A function f : X ! Y is said to be fuzzy h-irresolute if the inverse image of every h-open fuzzy set in Y is h-open fuzzy set in X. Lemma 26. Let f : X ! Y be a function, then the following are equivalent: (1) f is fuzzy h-irresolute, (2) f(h-cl(c)) 6 h-cl(f(c)) for every fuzzy set c in X. Theorem 27. Let f : X ! Y be a fuzzy h-irresolute surjection function. If X is a h-closed space, then Y is h-closed. Proof. Let {gj : j 2 J} be a h-open fuzzy set cover of Y. Then {f1(gj) : j 2 J} is a h-open fuzzy set cover of X. By hypothesis, there exists a finite subset F J such that _j2Fh-cl(f1(gj)) = 1X. From the surjectivity of f and by the previous lemma, 1Y ¼ f ð1X Þ ¼ f ð_j2F h-clðf 1 ðgj ÞÞÞ 6 _j2F h-clðf ðf 1 ðgj ÞÞÞ ¼ _j2F h-clðgj Þ. Hence, Y is h-closed.
h
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Caldas M, Jafari S. h-Compact fuzzy topological spaces. Chaos, Solitons & Fractals 2005;25:229–32. Chang CL. Fuzzy topological spaces. J Math Anal Appl 1968;24:182–90. Di Concillio A, Gerla G. Almost compactness in fuzzy topological spaces. Fuzzy Set Syst 1984;13:187–92. El-Naschie MS. On the uncertainty of cantorian geometry and the two-slit experiment. Chaos, Solitons & Fractals 1998;9(3):517–29. El-Naschie MS. On the certification of heterotic strings, theory and 1 theory. Chaos, Solitons & Fractals 2000:2397–408. Maio GD, Noiri T. On s-closed spaces. Indian J Pure Appl Math 1987;18(3):226–33. Mukherjee MN, Sinha SP. On some near-fuzzy continuous functions between fuzzy topological spaces. Fuzzy Set Syst 1990;34:245–54. Mukherjee MN, Sinha SP. Fuzzy h-closure operator on fuzzy topological spaces. Int J Math Math Sci 1991;14:309–14. Nanda S. Strongly compact fuzzy topological spaces. Fuzzy Set Syst 1991;42:259–62. Pu P-M, Liu Y-M. Fuzzy topology I. Neighborhood structure of a fuzzy point and Moore–Smith convergence. J Math Anal Appl 1980;76:571–99. Zahran AM. Strongly compact and P-closed fuzzy topological spaces. J Fuzzy Math 1995;3(1):97–102.
More on h-compact fuzzy topological spaces Erdal Ekici Department of Mathematics, Canakkale Onsekiz Mart University, Terzioglu Campus, 17020 Canakkale, Turkey Accepted 8 April 2005
Abstract Recently, El-Naschie has shown that the notion of fuzzy topology may be relevant to quantum particle physics in connection with string theory and e1 theory. In 2005, Caldas and Jafari have introduced h-compact fuzzy topological spaces. The purpose of this paper is to investigate further properties of h-compact fuzzy topological spaces. Moreover, the notion of h-closed fuzzy topological spaces is introduced and properties of it are obtained. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction In the literature there are many papers on compactness. Compactness occupies a very important place in fuzzy topology and so do some of its forms [3,6,9,11], etc. Moreover, El-Naschie [4,5] has shown that the notion of fuzzy topology may be relevant to quantum particle physics in connection with string theory and e1 theory. Also, applicability of the new concepts in this paper to superstrings and e1 space time could be probably possible in the near future. In [7,8], Mukherjee and Sinha introduced the concept of h-open sets in fuzzy setting. In [1], the notion of h-compactness is introduced by Caldas and Jafari. The aim of this paper is devoted to introduce the further properties of h-compact fuzzy topological space and to introduce h-closed fuzzy topological space. The notion of h-closedness comes out to be strictly weaker than fuzzy almost compactness. Also, the relationships between fuzzy h-compactness and fuzzy filterbases and the relationships between fuzzy filterbases and fuzzy h-closedness are obtained.
2. Preliminaries Throughout this paper X and Y mean fuzzy topological spaces. A fuzzy point xp in X is a fuzzy set having support x 2 X and value p 2 (0, 1] [10]. The complement and the support of a fuzzy set c denoted by cc and S(c), respectively. For two fuzzy sets c and q, we shall write cqqðcqqÞ to mean that c is quasi coincident (not quasi coincident) with q, i.e, there exists x 2 X such that c(x) + q(x) > 1 (c(x) + q(x) 6 1) [10]. Definition 1 ([7,8]). A fuzzy point xp in a fuzzy topological space X is said to be a fuzzy h-cluster point of a fuzzy set k if and only if for every fuzzy open q-neighbourhood l of xp, cl(l) is q-coincident with k. The set of all fuzzy h-cluster E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.077
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points of k is called the fuzzy h-closure of k and is denoted by h-cl(k). A fuzzy set k is fuzzy h-closed if and only if k = h-cl(k). The complement of a fuzzy h-closed set is called fuzzy h-open. The h-interior of k denoted by h-int(k) is defined as h-intðkÞ ¼ fxp : for some fuzzy open q-neighbourhood b of xp ; clðbÞ 6 kg. Definition 2. A fuzzy topological space X is said to be almost compact [3] if and only if every open cover of X has a finite subcollection whose closures cover X. Lemma 3. Let q be a fuzzy set in a fuzzy topological space X, then (1) (2) (3) (4) (5)
q is a fuzzy h-open if and only if q = h-int(q), Xnh-int(q) = h-cl(Xnq), h-int(Xnq) = Xnh-cl(q), h-cl(q) is a fuzzy closed set but not necessarily a fuzzy h-closed set, h-int(q) is a fuzzy open set but not necessarily a fuzzy h-open set.
Remark 4. Let q be a fuzzy set in a fuzzy topological space X, then (1) (2) (3) (4)
cl(q) 6 h-cl(q), h-int(q) 6 int(q), if q is fuzzy open set, then cl(q) = h-cl(q), if q is fuzzy closed set, then int(q) = h-int(q).
Definition 5 [2]. A fuzzy topological space (X, s) is called an extremally disconnected space if cl(g) 2 s for every fuzzy open set g.
3. Properties of fuzzy h-compact space In this section, further properties of fuzzy h-compact spaces are investigated. Definition 6. A collection W of fuzzy sets in a fuzzy topological space X is said to be cover of a fuzzy set c of X if and only if (_g2W)(x) = 1X, for every x 2 S(c). A fuzzy cover W of a fuzzy set c in a fuzzy topological space X is said to have a finite subcover if and only if there exists a finite subcollection D = {g1, . . ., gn} of W such that ð_nj¼1 gj ÞðxÞ P cðxÞ, for every x 2 S(c). Definition 7 [1]. A fuzzy topological space X is said to be fuzzy h-compact if and only if for every family W of h-open fuzzy sets such that _g2Wg = 1X there exists a finite subfamily D W such that _g2Dg = 1X. Definition 8 [1]. A fuzzy set c in a fuzzy topological space X is said to be fuzzy h-compact relative to X if and only if for every family W of h-open fuzzy sets such that _g2Wg P c(x) there exists a finite subfamily D W such that _g2Dg P c(x) for every x 2 S(c). Definition 9. A collection of fuzzy subsets n of a fuzzy topological space X is said to form a fuzzy filterbases if and only if for every finite collection {gj : j = 1, . . ., n}, ^nj¼1 gj 6¼ 0X . Theorem 10. A fuzzy topological space X is fuzzy h-compact if for every filterbases n in X, ^l2nh-cl(l) 5 0X. Proof. Let W be a h-open fuzzy set cover of X and W has no a finite subcover. Then for every finite subcollection {g1, . . ., gn} of W, there exists x 2 X such that gj(x) < 1 for every j = 1, . . ., n. Then gcj ðxÞ > 0, so that ^nj¼1 gcj ðxÞ 6¼ 0X . Thus, fgcj ðxÞ : gj 2 Wg forms a filterbases in X. Since W is h-open fuzzy set cover of X, then ð_gj 2W gj ÞðxÞ ¼ 1X for every x 2 X and hence ^gj 2W h-clðgci ðxÞÞ ¼ ^gj 2W gcj ðxÞ ¼ 0X ;
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which is a contradiction. Then every h-open fuzzy set cover of X has a finite subcover and hence X is fuzzy hcompact. h Theorem 11. If a fuzzy extremally disconnected topological space X is fuzzy h-compact, then for every filterbases n in X, ^l2nh-cl(l) 5 0X. Proof. Suppose there exists of filterbases n such that ^l2nh-cl(l) = 0X, so that (_l2n(h-cl(l))c)(x) = 1X for every x 2 X and hence W ¼ fðh-cllÞc : l 2 ng is a h-open fuzzy set cover of X. Since X is fuzzy h-compact, then W has a finite subcover. Then ð_nj¼1 ðh-clðlj ÞÞc ÞðxÞ ¼ 1X and hence ð_nj¼1 lcj ÞðxÞ ¼ 1X , so that ^nj¼1 lj ¼ 0X which is contradiction, since the lj are members of filterbases n. Therefore ^l2nh-cl(l) 5 0X for every filterbases n. h Corollary 12. Let X be a fuzzy extremally disconnected topological space. Then X is fuzzy h-compact if and only if for every filterbases n in X, ^l2nh-cl(l) 5 0X. Theorem 13. A fuzzy set c in a fuzzy topological space X is fuzzy h-compact relative to X if for every filterbases n such that every finite of members of n is quasi coincident with c, (^l2nh-cl(l)) ^ c 5 0X. Proof. Let c not be fuzzy h-compact relative to X, then there exists a h-open fuzzy set W cover of c such that W has no finite subcover D. Then ð_gj 2D gj ÞðxÞ < cðxÞ for some x 2 S(c), so that ð^nj 2D gcj ÞðxÞ > cc ðxÞ P 0 and hence n ¼ fgcj : gj 2 Wg forms a filterbases and ^gj 2D gcj qc. By hypothesis ð^gj 2D h-clðgcj ÞÞ ^ c 6¼ 0X and hence ð^gj 2D gcj Þ ^ c 6¼ 0X . Then for some x 2 S(c), ð^gj 2W gcj ÞðxÞ > 0X , that is ð_gj 2W gj ÞðxÞ < 1X , which is a contradiction. Hence c is fuzzy h-compact relative to X. h Theorem 14. If a fuzzy set c in a fuzzy extremally disconnected topological space X is fuzzy h-compact relative to X, then for every filterbases n such that every finite of members of n is quasi coincident with c, (^l2nh-cl(l)) ^ c 5 0X. Proof. Suppose that there exists a filterbases n such that every finite of members of n is quasi coincident with c and (^l2nh-cl(l)) ^ c = 0X. Then for every x 2 S(c), (^l2nh-cl(l))(x) = 0X and hence (_l2n(h-cl(l))c)(x) = 1X for every x 2 S(c). Thus W = {(h-cll)c : l 2 n} is h-open fuzzy set cover of c. Since c is fuzzy h-compact relative to X, then there exists a finite subcover, say fðh-clðl1 ÞÞc ; . . . ; ðh-clðln ÞÞc g; such that ð_nj¼1 ðh-clðlj ÞÞc ÞðxÞ P cðxÞ for every x 2 S(c). Hence ð^nj¼1 h-clðlj ÞÞðxÞ 6 cc ðxÞ for every x 2 S(c), so that ^nj¼1 ðh-clðlj ÞÞqc, which is a contradiction. Therefore, for every filterbases n such that every finite of members of n is quasi coincident with c, (^l2nh-cl(l)) ^ c 5 0X. h Corollary 15. Let X be a fuzzy extremally disconnected topological space. Then a fuzzy set c in X is fuzzy h-compact relative to X if and only if for every filterbases n such that every finite of members of n is quasi coincident with c, (^l2nhcl(l)) ^ c 5 0X. Definition 16. A function f : X ! Y is said to be fuzzy h-open if and only if the image of every h-open fuzzy set in X is hopen in Y. Theorem 17. Let f : X ! Y be a fuzzy h-open bijective function and Y is fuzzy h-compact, then X is fuzzy h-compact. Proof. Let {gj : j 2 J} be a collection of h-open fuzzy set cover of X, then {f(gj) : j 2 J} is h-open fuzzy set covering of Y. Since Y is fuzzy h-compact, there exists a finite subset F J such that {f(gj) : j 2 F} is a cover of Y. We have 1X ¼ f 1 ð1Y Þ ¼ f 1 ðf ð_j2F gj ÞÞ ¼ _j2F gj and therefore, X is fuzzy h-compact.
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4. A new form of fuzzy compactness and properties of this form In this section, the concept of fuzzy h-closed topological spaces is introduced and properties of this notion are investigated. Definition 18. A fuzzy set c in a fuzzy topological space X is said to be a h-q-neighbourhood of a fuzzy point xp in X if there exists a h-open fuzzy set g 6 c such that xpqg. Theorem 19. Let xp be a fuzzy point in a fuzzy topological space X and c be any fuzzy set of X, then xp 2 h-cl(c) if and only if for every h-q-neighbourhood q of xp, qqc. qc. Then there exists a h-open fuzzy set Proof. ()): Let xp 2 h-cl(c) and there exists a h-q-neighbourhood q of xp, q g 6 q in X such that xpqg, which implies gqc and hence c 6 gc. Then h-cl(c) 6 gc. Since xp 62 gc, then xp 62 h-cl(c), which is a contradiction. ((): Obvious. h Definition 20. A fuzzy topological space X is said to be h-closed if and only if for every family W of h-open fuzzy set such that _g2Wg = 1X there exists a finite subfamily D W such that (_g2Dh-cl(g))(x) = 1X, for every x 2 X. Remark 21. The following implication holds for a fuzzy topological space X: fuzzy almost compact ) fuzzy h-closed. Question: Is there an example showing that fuzzy h-closedness does not imply fuzzy almost compactness? Theorem 22. A fuzzy extremally disconnected topological space X is h-closed if and only if for every fuzzy h-open filterbases n in X, ^l2nh-cl(l) 5 0X. Proof. Let W be a h-open fuzzy set cover of X and let for every finite subfamily D of W, (_g2Dh-cl(g))(x) < 1X for some x 2 X. Then (^g2Dh-cl(gc))(x) > 0X for some x 2 X. Thus, fðh-clðgÞÞc : g 2 Wg ¼ n forms a fuzzy h-open filterbases in X. Since W is a h-open fuzzy set cover of X, then ^g2Wgc = 0X which implies ^g2Whcl(h-cl(g)c) = 0X, which is a contradiction. Then every h-open fuzzy set W cover of X has a finite subfamily D such that (_g2Dh-cl(g))(x) = 1X for every x 2 X. Hence, X is h-closed. Conversely, suppose there exists a fuzzy h-open filterbases n in X such that ^l2nh-cl(l) = 0X, so that (_l2n(hcl(l))c)(x) = 1X for every x 2 X and hence W = {(h-cll)c : l 2 n} is a h-open fuzzy set cover of X. Since X is h-closed, then W has a finite subfamily D such that ð_l2D h-clððh-clðlÞÞc ÞÞðxÞ ¼ 1X for every x 2 X, and hence ^l2D(h-cl((h-cl(l))c))c = 0X. Thus, ^l2Dl = 0X which is a contradiction, since all the l are members of filterbases. h Definition 23. A fuzzy set c in a fuzzy topological space X is said to be h-closed relative to X if and only if for every family W of h-open fuzzy sets such that _g2Wg P c, there exists a finite subfamily D W such that (_g2Dhcl(g))(x) P c(x) for every x 2 S(c). Theorem 24. A fuzzy subset c in a fuzzy extremally disconnected topological space X is h-closed relative to X if and only if every fuzzy h-open filterbases n in X, (^l2nh-cl(l)) ^ c = 0X, there exists a finite subfamily k of n such that ð^l2k lÞ qc. Proof. Let c be a h-closed relative to X, suppose n is a fuzzy h-open filterbases in X such that for every finite subfamily k of n, (^l2kl)qc, but (^l2nh-cl(l)) ^ c = 0X. Then for every x 2 S(c), (^l2nh-cl(l))(x) = 0X and hence (_l2n(hcl(l))c)(x) = 1X for every x 2 S(c). Then W = {(h-cll)c : l 2 n} is a h-open fuzzy set cover of c and hence there exists a finite subfamily k n such that (_l2kh-cl(h-cl(l))c) P c, so that ^l2k ðh-clðh-clðlÞÞc Þc ¼ ^l2k h-intðh-clðlÞÞ 6 cc and hence ^l2kl 6 cc. Then ^l2k lqc which is a contradiction.
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Conversely, let c not be a h-closed fuzzy set relative to X, then there exists a h-open fuzzy set W cover of c such that every finite subfamily D W, ð_l2D h-clðgÞÞðxÞ 6 cðxÞ for some x 2 S(c) and hence (^l2D(h-cl(g))c)(x) > cc(x) P 0 for some x 2 S(c). Thus n = {(h-clg)c : g 2 W} forms a fuzzy h-open filterbases in X. Let there exists a finite subfamily {(hcl(g))c : g 2 D} such that ð^g2D ðh-clðgÞÞc Þqc. Then c 6 _g2Dh-cl(g). So there exists a finite subfamily D W such that _g2Dh-cl(g) P c which is a contradiction. Then for each finite subfamily k = {(h-clg)c : g 2 D} of n, We have (^g2D(hclg)c)qc. Hence by the given condition (^g2Wh-cl((h-cl(g))c)) ^ c 5 0X, so there exists x 2 S(c) such that (^g2Wh-cl((hcl(g))c))(x) > 0X. Then ð_g2W ðh-clððh-clðgÞÞc ÞÞc ÞðxÞ ¼ ð_g2W h-intðh-clðgÞÞÞðxÞ < 1X and hence, (_g2Wg)(x) < 1X which contradicts the fact that W is a h-open fuzzy set cover of c. Therefore, c is fuzzy h-closed relative to X. h Definition 25. A function f : X ! Y is said to be fuzzy h-irresolute if the inverse image of every h-open fuzzy set in Y is h-open fuzzy set in X. Lemma 26. Let f : X ! Y be a function, then the following are equivalent: (1) f is fuzzy h-irresolute, (2) f(h-cl(c)) 6 h-cl(f(c)) for every fuzzy set c in X. Theorem 27. Let f : X ! Y be a fuzzy h-irresolute surjection function. If X is a h-closed space, then Y is h-closed. Proof. Let {gj : j 2 J} be a h-open fuzzy set cover of Y. Then {f1(gj) : j 2 J} is a h-open fuzzy set cover of X. By hypothesis, there exists a finite subset F J such that _j2Fh-cl(f1(gj)) = 1X. From the surjectivity of f and by the previous lemma, 1Y ¼ f ð1X Þ ¼ f ð_j2F h-clðf 1 ðgj ÞÞÞ 6 _j2F h-clðf ðf 1 ðgj ÞÞÞ ¼ _j2F h-clðgj Þ. Hence, Y is h-closed.
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