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On fuzzy upper and lower s-limit sets
Chaos, Solitons and Fractals 28 (2006) 1090–1098 www.elsevier.com/locate/chaos
On fuzzy upper and lower s-limit sets Erdal Ekici Department of Mathematics, Canakkale Onsekiz Mart University, Terzioglu Campus, 17020 Canakkale, Turkey Accepted 10 August 2005
Abstract The aim of this paper is to introduce the notions of fuzzy upper and lower s-limits sets. Properties and basic relationships among fuzzy upper s-limit set, fuzzy lower s-limit set, fuzzy semi-continuity are investigated. Possible applications to quantum physics and computer graphics are touched upon. We speculate that s-convergence and s-properties may be relavant to the physics of fractal and cantorian spacetime. 2005 Elsevier Ltd. All rights reserved.
1. Introduction This paper will give some new topological properties (for example s-limit sets, s-convergence, s-continuously convergence, s-separation properties) which have been found to be very useful in the study of certain objects of digital topology. Thus we may stress once more the importance of s-convergence as a branch of them and the possible applications in quantum physics [4–7] and computer graphics [8–10]. We speculate that s-convergence and s-properties may be relavant to the physics of fractal and cantorian spacetime [4,6,7,13]. In this paper, the symbol I will denote the unit interval [0, 1]. A fuzzy set in X is a function with domain X and values in I, that is, an element of IX. Let l 2 IX. The subset of X in which l assumes nonzero values, is known as the support of l [18]. A member l of IX is contained in a member q of IY denoted l 6 q if and only if l(x) 6 q(x), for every x 2 X [18]. By l · q we denote the fuzzy set in X · Y for which (l · q)(x, y) = min{l(x),q (y)}, for every (x, y) 2 X · Y. For l 2 IX, the fuzzy set lc is defined by lc(x) = 1 l(x) for every x 2 X. A fuzzy set in X is called a fuzzy point if and only if it takes the value 0 for all y 2 X except one, say, x 2 X. If its value at x is e (0 < e 6 1) we denote the fuzzy point by pex , where the point x is called its support. The class of all fuzzy points in X is denoted by v [11,14]. The fuzzy point pex is said to be contained in a fuzzy set l or to belong to l, denoted by pex 2 l, if and only if e 6 l(x). Chang [2] introduced the concept of fuzzy topology. A fuzzy set l in a fuzzy topological space (X, s) is called a neighbourhood of a fuzzy point pex if and only if there exists a f 2 s such that pex 2 f 6 l [11]. A fuzzy point pex is said to be quasi-coincident with l denoted by pex ql if and only if e > lc(x) or e + l(x) > 1 [11]. A fuzzy set l is said to be quasi-coincident with q, denoted lqq, if and only if there exists x 2 X such that l(x) > qc(x) or l(x) + q(x) > 1 [11]. If l does not quasi-coincident with q, then we write l qq. A fuzzy set l of a fuzzy topological space (X, s) is called a fuzzy semi-open set if l 6 cl(int(l)) [1]. Also, a fuzzy set l of a fuzzy topological space (X, s) is called fuzzy semi-closed if the fuzzy set lc is fuzzy semi-open. Finally, by s-cl(l)
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and s-int(l) we denote the fuzzy set: s-cl(l) = ^{q 2 IX : q is fuzzy semi-closed, l 6 q}, s-int(l) = _{q 2 IX : q is fuzzy semi-open, q 6 l}, respectively [17]. The family of all fuzzy semi-open sets of X is denoted by FSO(X). A fuzzy set l in a fuzzy topological space (X, s) is called a Q-neighbourhood of pex if and only if there exists q 2 s such that pex qq and q 6 l. In what follows by Nðpex Þ we denote the family of all fuzzy open Q-neighbourhoods of the fuzzy point pex in X. A fuzzy point pex 2 clðlÞ if and only if each Q-neighbourhood of pex is quasi-coincident with l [11]. A fuzzy set l in a fuzzy space X is called a fuzzy semi-neighborhood of a fuzzy point pex if there exists a g 2 FSO(X) such that pex 2 g 6 l. A fuzzy set l in a fuzzy space X is called a fuzzy Q-s-neighborhood of pex if there exists q 2 FSO(X) such that pex qq and q 6 l. The set N s ðpex Þ with the relation 6* (that is, n1 6 *n2 if and only if n2 6 n1) form a directed set. The basic knowledges related with the fuzzy functions can be seen in [2,3,12,15,16].
2. Fuzzy s-continuously converge Definition 1. A function f from a fuzzy topological space Y into a fuzzy topological space Z is called fuzzy semicontinuous if and only if for every fuzzy point pex in Y and every fuzzy semi-open Q-neighbourhood f of f ðpex Þ, there exists a fuzzy semi-open Q-neighbourhood n of pex such that f(n) 6 f. Let Y and Z be fuzzy topological spaces. By s-FC(Y, Z) we denote the set of all fuzzy semi-continuous functions of Y into Z. Definition 2. Let (X, s) be a fuzzy topological space and let {pi, i 2 I} be a net of fuzzy points in X. We say that the fuzzy net {pi, i 2 I} fuzzy s-converges to a fuzzy point p of X if for every fuzzy semi-open Q-neighbourhood n of p in X there exists i0 2 I such that piqn, for every i 2 I and i P i0. Theorem 3. Let l be a fuzzy set of a fuzzy space X. Then, a fuzzy point pex 2 s-clðlÞ if and only if for every n 2 FSO(X) for which pex qn we have nql. Proof. The fuzzy point pex 2 s-clðlÞ if and only if pex 2 f, for every fuzzy semi-closed set f of X for which l 6 f. Equivalently pex 2 s-clðlÞ if and only if e 6 1 n(x), for every fuzzy semi-open set n for which l 6 1 n. Thus pex 2 s-clðlÞ if and only if n(x) 6 1 e, for every fuzzy semi-open set n for which n 6 1 l. So, pex 2 s-clðlÞ if and only if for every fuzzy semi-open set n of X such that n(x) > 1 e we have n i X l. Therefore, pex 2 s-clðlÞ if and only if for every fuzzy semi-open set n of X such that n(x) + e > 1 we have nql. Thus, pex 2 s-clðlÞ if and only if for every fuzzy semi-open set n of X such that pex qn we have nql. h Theorem 4. Let f : Y ! Z be a fuzzy semi-continuous map, p be a fuzzy point in Y and n, f be fuzzy semi-open Q-neighqf. bourhoods of p and f(p), respectively such that f(n) i f. Then there exists a fuzzy point p1 in Y such that p1qn and f ðp1 Þ Proof. Since f(n) i f, we have n i f1(f). Thus there exists x 2 Y such that n(x) > f1(f)(x) or n(x) f1(f)(x) > 0 and therefore n(x) + 1 f1(f)(x) > 1 or n(x) + (f1(f))c(x) > 1. Let (f1(f))c(x) = r. Clearly, for the fuzzy point prx we have prx qn and prx 2 ðf 1 ðfÞÞc . Hence for the fuzzy point p1 ¼ prx we have p1qn and f ðp1 Þ qf. h Theorem 5. A map f from a fuzzy topological space Y into a fuzzy topological space Z is fuzzy semi-continuous if and only if for every fuzzy point p of Y and for every net {pi, i 2 I} which fuzzy s-converges to p the net {f(pi), i 2 I} of Z fuzzy s-converges to f(p). Definition 6. A net {fj, j 2 J} in s-FC(Y, Z) fuzzy s-continuously converges to f 2 s-FC(Y, Z) if and only if for every fuzzy net {pi, i 2 I} in Y which fuzzy s-converges to a fuzzy point p in Y we have that the fuzzy net {fj(pi),(i, j) 2 I · J} fuzzy s-converges to the fuzzy point f(p) in Z. Theorem 7. The following properties hold: (1) If {fi, i 2 I} is a net in s-FC(Y, Z) such that fi = f, for every i 2 I, then the {fi, i 2 I} fuzzy s-continuously converges to f 2 s-FC(Y, Z). (2) If {fi, i 2 I} is a net in s-FC(Y, Z), which fuzzy s-continuously converges to f 2 s-FC(Y, Z) and {gj, j 2 J} be a subnet of {fi, i 2 I}, then the net {gj, j 2 J} fuzzy s-continuously converges to f. (3) If {fi, i 2 I} is a net in s-FC(Y, Z) which does not fuzzy s-continuously converges to f 2 s-FC(Y, Z), then there exists no subnet of {fi, i 2 I}, which fuzzy continuously converges to f.
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Theorem 8. A net {fj, j 2 J} in s-FC(Y, Z) fuzzy s-continuously converges to f 2 s-FC(Y, Z) if and only if for every fuzzy point p in Y and for every fuzzy semi-open Q-neighbourhood f of f(p) in Z there exist an element j0 2 J and a fuzzy semiopen Q-neighbourhood n of p in Y such that fj(n) 6 f, for every j P j0, j 2 J. Proof. Let p be a fuzzy point in Y and let f be a fuzzy semi-open Q-neighbourhood of f(p) in Z such that for every j 2 J and for every fuzzy semi-open Q-neighbourhood n of p in Y there exists j 0 P j such that fj0 ðnÞif. Then for every fuzzy semi-open Q-neighbourhood n of p in Y we can choose a fuzzy point pn in Y by Theorem 4 such that pnqn and fj0 ðpn Þ qf. Clearly, the fuzzy net {pn,n 2 N(p)} fuzzy s-converges to p, but the fuzzy net ffj ðpn Þ; ðn; jÞ 2 NðpÞ J g does not fuzzy s-converge to f(p) in Z. Conversely, let {pi, i 2 I} be a fuzzy net in s-FC(Y, Z) which s-converges to the fuzzy point p in Y and let f be an arbitrary fuzzy semi-open Q-neighbourhood of f(p) in Z. By assumption there exists a fuzzy semi-open Qneighbourhood n of p in Y and an element j0 2 J such that fj(n) 6 f, for every j P j0, j 2 J. Since the fuzzy net {pi, i 2 I} s-converges to p in Y. There exists i0 2 I such that piqn, for every i 2 I, i P i0. Let (i0, j0) 2 I · J. Then for every (i, j) 2 I · J, (i, j) P (i0, j0) we have fj(pi)qfj(n) and fj(n) 6 f, that is fj(pi)qf. Thus the net {fj(pi), (i, j) 2 I · J} fuzzy sconverges to f(p) and the net {fj, j 2 J} fuzzy s-continuously converges to f. h Definition 9. A fuzzy set l of a fuzzy space X is called semi-generalized closed (briefly f-sg-closed) if s-cl(l) 6 n whenever l 6 n and n fuzzy semi-open set of X. Definition 10. A fuzzy space X is called s-T1 if every fuzzy point is fuzzy semi-closed. Theorem 11. A fuzzy space X is s-T1 if and only if for each x 2 X and each e 2 [0, 1] there exists a fuzzy semi-open set l such that l(x) = 1 e and l(y) = 1 for y 5 x. Proof. ()) Let e = 0. We set l = X. Then l is fuzzy semi-open set such that l(x) = 1 0 and l(y) = 1 for y 5 x. Now, let e 2 (0, 1] and x 2 X. We set l ¼ ðpex Þc . The set l is fuzzy semi-open such that l(x) = 1 e and l(y) = 1 for y 5 x. (() Let pex be an arbitrary fuzzy point of X. We prove that the fuzzy point pex is fuzzy semi-closed. By assumption there exists a fuzzy semi-open set l such that l(x) = 1 e and l(y) = 1 for y 5 x. Clearly, lc ¼ pex . Thus the fuzzy point pex is fuzzy semi-closed and therefore the fuzzy space X is s-T1. h Definition 12. A fuzzy space X is called a quasi-s-T1 if for any fuzzy points pex and pay for which suppðpex Þ ¼ x 6¼ suppðpay Þ ¼ y, there exists a fuzzy semi-open set n such that pex 2 n and pay 62 n and another g such that pex 62 g and pay 2 g. 0;75 Example 13. Let (X, s) be a fuzzy topological space such that X = {x, y} and s ¼ f£; X ; p0;30 x ; p x g. The fuzzy topological space X is quasi-s-T1 but it is not s-T1.
Definition 14. A fuzzy space X is called a s-T2 space if for any fuzzy points pex and pay for which suppðpex Þ ¼ x 6¼ suppðpay Þ ¼ y, there exist two fuzzy semi-open Q-s-neighbourhoods n and g of pex and pay , respectively, such that n ^ g = B. Definition 15. A fuzzy point pex is called weak (respectively, strong) if e 6 12 (respectively, e > 12) [11]. Theorem 16. If X is a quasi-s-T1 fuzzy space and pex a weak fuzzy point in X, then ðpex Þc is a fuzzy semi-neighborhood of each fuzzy point pay with y 5 x. Proof. Let y 5 x and pay be a fuzzy point of X. Since the space X is a quasi-s-T1 there exists a fuzzy semi-open n of X such that pay 2 n and pex 62 n. This implies that e > n(x). Also, e 6 12. Thus n(x) 6 1 e. Therefore nðyÞ 6 1 ¼ ðpex Þc ðyÞ; for every y 2 Xn{x}. So n 6 ðpex Þc . Hence, the fuzzy point pex is a semi-neighborhood of pay .
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Definition 17. A fuzzy space X is called a s-regular space if for any fuzzy point pex and a fuzzy semi-closed set f not containing pex , there exist n, g 2 FSO(X) such that pex 2 n, f 6 g and n ^ g = B. Theorem 18. If X is a s-regular fuzzy space, then for any strong fuzzy point pex and any fuzzy semi-open set n containing pex , there exists a fuzzy semi-open set q containing pex such that s-cl(q) 6 n. Proof. Suppose that pex is any strong fuzzy point contained in n 2 FSO(X). Then 12 < e 6 nðxÞ. Thus the complement of n, that is the fuzzy set nc, is a fuzzy semi-closed set to which does not belong the fuzzy point pex . Thus, there exist q, g 2 FSO(X) such that pex 2 q and nc 6 g with q ^ g = B. Hence, we have q 6 gc and s-clðqÞ 6 s-clðgc Þ ¼ gc . Now nc 6 g implies gc 6 n. This means that s-cl(q) 6 n which complete the proof.
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Theorem 19. If X is a fuzzy s-regular space, then the strong fuzzy points in X are f-sg-closed. Proof. Let pex be any strong fuzzy point in X and n be a fuzzy semi-open set such that pex 2 n. By the previous theorem there exists a q 2 FSO(X) such that pex 2 q and s-cl(q) 6 n. We have s-clðpex Þ 6 s-clðqÞ 6 n. Hence, the fuzzy point pex is f-sg-closed. h Definition 20. A fuzzy space X is called a weakly s-regular space if for any weak fuzzy point pex and a fuzzy semi-closed set f not containing pex , there exist n, g 2 FSO(X) such that pex 2 n, f 6 g and n ^ g = B. Definition 21. A fuzzy set n in a fuzzy space X is said to be fuzzy s-nearly crisp if s-cl(n) ^ (s-cl(n))c = B. Theorem 22. Let X be a fuzzy space. If for any weak fuzzy point pex and any n 2 FSO(X) containing pex , there exists a fuzzy semi-open and s-nearly crisp fuzzy set q containing pex such that s-cl(q) 6 n, then X is fuzzy weakly s-regular. Proof. Assume that f is a fuzzy semi-closed set not containing the weak fuzzy point pex . Then fc is a fuzzy semi-open set containing pex . By hypothesis, there exists a fuzzy semi-open and s-nearly crisp fuzzy set q such that pex 2 q and s-cl(q) 6 fc. We set g = s-int(s-cl(q)) and l = 1 s-cl(q). Then g is fuzzy semi-open, pex 2 g and f 6 l. We are going to prove that l ^ g = B. Now assume that there exists y 2 X such that (g ^ l)(y) = a 5 B. Then pay 2 g ^ l. Hence, pay 2 s-clðqÞ and pay 2 ðs-clðqÞÞc . This is a contradiction since q is s-nearly crisp. Hence, the fuzzy space X is weakly s-regular. h Definition 23. Let l be a fuzzy set of a fuzzy space X. A fuzzy point pex is called a semi-boundary point of a fuzzy set l if and only if pex 2 s-clðlÞ ^ ðX s-clðlÞÞ. By s-bd(l) we denote the fuzzy set s-cl(l) ^ (X s-cl(l)). Theorem 24. Let X be a fuzzy space. Suppose that pex and pay be weak and strong fuzzy points, respectively. If pex is semigeneralized closed, then pay 2 s-clðpex Þ ) pex 2 s-clðpay Þ. Proof. Suppose that pay 2 s-clðpex Þ and pex 62 s-clðpay Þ. Then s-clðpay ÞðxÞ < e. Also e 6 12. Thus s-clðpay ÞðxÞ 6 1 e and therefore e 6 1 s-clðpay ÞðxÞ. So pex 2 ðs-clðpay ÞÞc . But pex is semi-generalized closed and ðs-clðpay ÞÞc fuzzy semi-open. Hence, s-clðpex Þ 6 ðs-clðpay ÞÞc . By assumption we have pay 2 s-clðpex Þ. Thus, pay 2 ðs-clðpay ÞÞc . We prove that this is a contradiction. Indeed, we have a 6 1 s-clðpay ÞðyÞ or s-clðpay ÞðyÞ 6 1 a. Also pay 2 s-clðpay Þ. Thus a 6 1 a. But pay is a strong fuzzy point, that is a > 12. So the above relation a 6 1 a is a contradiction. Hence, pex 2 s-clðpay Þ. h Theorem 25. Let l be a fuzzy set of a fuzzy space X. Then l _ s-bd(l) 6 s-cl(l}. Proof. Let pex 2 l _ s-bdðlÞ. Then pex 2 l or pex 2 s-bdðlÞ. If pex 2 s-bdðlÞ, then pex 2 s-clðlÞ. Let us suppose that pex 2 l. We have s-cl(l) = ^{f : f 2 IX, f is semi-closed and l 6 f}. So, if pex 2 l, then pex 2 f, for every fuzzy semi-closed f of X for which l 6 f and therefore pex 2 s-clðlÞ. h
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Definition 26. A fuzzy point pex in a fuzzy space X is said to be (1) well-semi-closed if there exists pay 2 s-clðpex Þ such that suppðpex Þ 6¼ suppðpay Þ, (2) just-semi-closed if the fuzzy set s-clðpex Þ is again fuzzy point. Clearly, in a fuzzy s-T1 space every fuzzy point is just-semi-closed. Theorem 27. If X is a fuzzy space and pex is a f-sg-closed but and well semi-closed fuzzy point, then X is not quasi-s-T1 space. Proof. Let X be a fuzzy quasi-s-T1 space. By the fact pex is well-semi-closed, there exists a fuzzy point pay with suppðpex Þ 6¼ suppðpay Þ such that pay 2 s-clðpex Þ. Then there exists an n 2 FSO(X) such that pex 2 n and pay 62 n. Therefore s-clðpex Þ 6 n and pay 2 n. But this is a contradiction and hence X cannot be quasi-s-T1 space. h Theorem 28. Let X be a fuzzy space. If pex and pax are two fuzzy points such that e < a and pax is fuzzy semi-open, then pex is just-semi-closed if it is f-sg-closed. Proof. We prove that the fuzzy set s-clðpex Þ is again a fuzzy point. We have pex 2 pax and the fuzzy set pax is fuzzy semiopen. Since pex is f-sg-closed we have s-clðpex Þ 6 pax . Thus s-clðpex ÞðxÞ 6 a and s-clðpex ÞðzÞ 6 0, for every z 2 Xn{x}. So the fuzzy set s-clðpex Þ is a fuzzy point. h 3. Fuzzy upper and lower s-limit sets Definition 29. Let I be a directed set. Let X be an ordinary set. Let v be the collection of all fuzzy points in X. The function S : I ! v is called a fuzzy net in X. For every i 2 I, S(I) is often denoted by si and hence a net S is often denoted by {si, i 2 I} [11]. Definition 30. Let S = {si, i 2 I} be a fuzzy net in X. S is said to be quasi-coincident with l if for each i 2 I, si is quasicoincident with l. A fuzzy net {gj, j 2 J} in X is called a fuzzy subnet of a fuzzy net {si, i 2 I} in X if there is a map N:J ! I such that (a) gj = sN(j) and (b) for the element i0 2 I there is j0 2 J such that if j P j0, j 2 J, then N(j) P i0. Definition 31. Let {li, i 2 I} be a net of fuzzy sets in a fuzzy topological space Y. Then s-F-limI ðli Þ, we denote fuzzy upper s-limit of the net {li, i 2 I} in IY, that is, the fuzzy set which is the union of all fuzzy points prx in Y such that for every i0 2 I and for every fuzzy semi-open Q-neighbourhood n of prx in Y there exists an element i 2 I for which i P i0 and liqn. In other cases we set s-F-limI ðli Þ ¼ £. Theorem 32. Let {li, i 2 I} and {qi, i 2 I} be two nets of fuzzy sets in Y. Then the following properties hold: (1) (2) (3) (4) (5) (6) (7) (8)
The fuzzy upper s-limit is semi-closed, s-F-limI ðli Þ ¼ s-F-limI ðs-clðli ÞÞ, If li = l for every i 2 I, then s-F-limI ðli Þ ¼ s-clðlÞ, The fuzzy upper s-limit is not affected by changing a finite number of the li, s-F-limI ðli Þ 6 s-clð_fli : i 2 IgÞ, If li 6 qi for every i2I, then s-F-limI ðli Þ 6 s-F-limI ðqi Þ, s-F-limI ðli _ qi Þ ¼ s-F-limI ðli Þ _ s-F-limI ðqi Þ, s-F-limI ðli ^ qi Þ 6 s-F-limI ðli Þ ^ s-F-limI ðqi Þ.
Proof. (1) It is sufficient to prove that s-clðs-F-limI ðli ÞÞ 6 s-F-limI ðli Þ. Let pry 2 s-clðs-F-limI ðli ÞÞ and let n be an arbitrary fuzzy semi-open Q-neighbourhood of pry . Then, we have nqs-F-limI ðli Þ. Hence, there exists an element y 0 2 Y such that nðy 0 Þ þ s-F-limI ðli Þðy 0 Þ > 1. Let s-F-limI ðli Þðy 0 Þ ¼ a. Then, for the fuzzy point pay 0 in Y we have pay 0 qn and pay 0 2 s-F-limI ðli Þ. Thus, for every element i0 2 I there exists i P i0, i 2 I such that liqn. This means that pry 2 s-F-limI ðli Þ. (2) Clearly, it is sufficient to prove that for every fuzzy semi-open set n the condition nqli is equivalent to nqs-cl(li). Let nqli. Then there exists an element y 2 Y such that n(y) + li(y) > 1. Since li 6 s-cl(li) we have n(y) + s-cl(li)(y) > 1 and therefore n qs-cl(li).
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Conversely, let nqs-cl(li). Then there exists an element y 2 Y such that n(y) + s-cl(li)(y) > 1. Let s-cl(li)(y) = r. Then pry 2 s-clðli Þ and the fuzzy semi-open set n is a fuzzy semi-open Q-neighbourhood of pry . Thus, nqli. (3), (4): Obvious. (5) Let pry 2 s-F-limI ðli Þ and let n be a fuzzy semi-open Q-neighbourhood of pry in Y. Then for every i0 2 I there exists i 2 I, i P i0 such that liqn and therefore _{li : i 2 I} qn. Thus, pry 2 s-clð_fli : i 2 IgÞ. (7) Clearly, li 6 li _ qi and qi 6 li _ qi, for every i 2 I. Hence, by (6), s-F-limðli Þ 6 s-F-limðli _ qi Þ I
I
and s-F-limðqi Þ 6 s-F-limðli _ qi Þ. I
I
Thus, s-F-limI ðli Þ _ s-F-limI ðqi Þ 6 s-F-limI ðli _ qi Þ. Conversely, let pry 2 s-F-limI ðli _ qi Þ. We prove that pry 2 s-F-limI ðli Þ _ s-F-limI ðqi Þ. Let us suppose that pry 62 s-F-limI ðli Þ _ s-F-limI ðqi Þ. Then pry 62 s-F-limI ðli Þ and pry 62 s-F-limI ðqi Þ. Hence, there exists a fuzzy semi-open qn1 , for every i 2 I, i P i1. Also, there exists a fuzzy semiQ-neighbourhood n1 of pry and an element i1 2 I such that li qn2 , for every i 2 I, i P i2. Let n = n1 _ n2 and let i0 2 I open Q-neighbourhood n2 of pry and an element i2 2 I such that qi such that i0 P i1 and i0 P i2. Then the fuzzy set n is a fuzzy semi-open Q-neighbourhood of pry and li _ qi qn, for every i 2 I, i P i0. Since pry 2 s-F-limI ðli _ qi Þ this is a contradiction. Thus, pry 2 s-F-limI ðli Þ _ s-F-limI ðqi Þ. The other proofs can be obtained by using the definitions.
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Definition 33. Let {li, i 2 I} be a net of fuzzy sets in a fuzzy topological space Y. Then by s-F-limI ðli Þ, we denote the fuzzy lower s-limit of the net {li, i 2 I} in IY, that is, the fuzzy set which is the union of all fuzzy points prx in Y such that for every fuzzy semi-open Q-neighbourhood n of prx in Y there exists an element i0 2 I such that liqn, for every i 2 I, i P i0. In other case we set s-F-limI ðli Þ ¼ £. Theorem 34. For the fuzzy upper and lower s-limit we have s-F-limI ðli Þ 6 s-F-limI ðli Þ. Proof. The proof of this theorem follows by the Definitions 31 and 33.
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Theorem 35. Let {li, i 2 I} be a net of fuzzy sets in Y such that li1 6 li2 if and only if i1 6 i2. Then s-clð_fli : i 2 IgÞ ¼ s-F-limI ðli Þ. Proof. Let pry 2 s-clð_fli : i 2 IgÞ and let n be a fuzzy semi-open Q-neighbourhood of pry in Y. Then nq _ {li : i 2 I}. Hence, there exists i0 2 I such that nqli0 . By assumption we have nqli, for every i 2 I, i P i0. Thus pry 2 s-F-limI ðli Þ. Conversely, let pry 2 s-F-limI ðli Þ and let n be an arbitrary fuzzy semi-open Q-neighbourhood of pry in Y. Then there exists an element i0 2 I such that nqli, for every i 2 I, i P i0. Hence nq _ {li : i 2 I} and therefore pry 2 s-clð_fli : i 2 IgÞ. h Theorem 36. Let {li, i 2 I} be a net of fuzzy sets in Y. Then we have s-F -limI ðli Þ ¼ ^fs-clð_fli : i P i0 gÞ : i0 2 Ig. Proof. Let pry 2 s-F-limI ðli Þ and let i0 2 I. We prove that pry 2 s-clð_fli : i P i0 gÞ. Let n be an arbitrary fuzzy semi-open Q-neighbourhood of pry in Y. Then, there exists i P i0, i 2 I such that nqli. Thus, nq _ {li : i P i0} and therefore pry 2 s-clð_fli : i P i0 gÞ. Conversely, let pry 2 ^fs-clð_fli : i P i0 gÞ : i0 2 Ig. Then, we have pry 2 s-clð_fli : i P i0 gÞ, for every i0 2 I. We prove that pry 2 s-F-limI ðli Þ. Let n be an arbitrary fuzzy semi-open Q-neighbourhood of pry in Y and let i0 2 I. Then nq _ {li : i P i0}. We prove that there exists i 2 I, i P i0 such that liqn. Let us suppose that nqli ,for every i 2 I, i P i0. Then, for every i 2 I, i P i0 and for every y 2 Y we have n(y) + li(y) 6 1 and therefore
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nðyÞ þ ð_fli : i P i0 gÞðyÞ 6 1; which is a contradiction. Hence, pry 2 s-F-limI ðli Þ.
h
Theorem 37. Let {li, i 2 I} be a net of fuzzy semi-closed sets in Y such that li1 6 li2 if and only if i2 6 i1. Then s-F-limI ðli Þ ¼ ^fli : i 2 Ig. Proof. Let pry 2 ^fli : i 2 Ig. Then pry 2 li or r 6 li(y) for every i 2 I. Let i0 2 I and n be a fuzzy semi-open Q-neighbourhood of pry , that is r + n(y) > 1. Then there exists i 2 I, i P i0 such that li(y) + n(y) P r + n(y) > 1. Hence, liqn and therefore pry 2 s-F-limI ðli Þ. Conversely, let pry 2 s-F-limI ðli Þ and let pry 62 ^fli : i 2 Ig. Then there exists i0 2 I such that pry 62 li0 , that is r > li0 ðyÞ. Let n ¼ ðli0 Þc . Then n is fuzzy semi-open Q-neighbourhood of pry and for every i P i0, n qli , which is a contradiction. h Theorem 38. Let {li, i 2 I} and {qi, i 2 I} be two nets of fuzzy sets in Y. Then the following properties hold: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
If li 6 qi for every i 2 I, then s-F-limI ðli Þ 6 s-F-limI ðqi Þ, s-F-limI ðli _ qi Þ P s-F-limI ðli Þ _ s-F-limI ðqi Þ, s-F-limI ðli ^ qi Þ 6 s-F-limI ðli Þ ^ s-F-limI ðqi Þ, The fuzzy lower s-limit is semi-closed, s-F-limI ðli Þ ¼ s-F-limI ðs clðli ÞÞ, If li = l for every i 2 I, then s-F-limI ðli Þ ¼ s-clðlÞ, The fuzzy lower s-limit is not affected by changing a finite number of the li, ^fli : i 2 Ig 6 s-F-limI ðli Þ, s-F-limI ðli Þ 6 s-clð_fli : i 2 IgÞ, _f^fli : i P i0 g : i0 2 Ig 6 s-F-limI ðli Þ.
Proof. (2) Let pry 2 s-F-limI ðli Þ _ s-F-limI ðqi Þ. Then either pry 2 s-F-limI ðli Þ or pry 2 s-F-limI ðqi Þ. Let pry 2 s-F-limI ðli Þ. Then for every fuzzy semi-open Q-neighbourhood n of pry in Y there exists an element i0 2 I such that liqn, for every i P i0, i 2 I. Also li 6 li _ qi. Thus (li _ qi)qn, for every i 2 I, i P i0 and therefore pry 2 s-F-limI ðli _ qi Þ. The other proofs can be obtained by using the definitions. (8) Let pry 2 ^fli : i 2 Ig. We prove that pry 2 s-F-limI ðli Þ. Let us suppose that pry 62 s-F-limI ðli Þ. Then there exists a qn. This means that fuzzy semi-open Q-neighbourhood n of pry such that for every i 2 I there exists i 0 P i for which li0 li0 ðxÞ þ nðxÞ 6 1, for every x 2 Y. Now, since pry 2 ^fli : i 2 Ig and n is a fuzzy semi-open Q-neighbourhood of pry we have that r 6 li(y), for every i 2 I and r + n(y) > 1. Thus, li(y) + n(y) > 1, for every i 2 I. By the above this is a contradiction. Hence, pry 2 s-F-limI ðli Þ. (10) Let pry 2 _f^fli : i P i0 g : i0 2 Ig. Then, there exists i0 2 I such that pry 2 ^fli : i P i0 g. Hence, pry 2 li , for every i 2 I, i P i0 and therefore r 6 li(y), for every i 2 I, i P i0. We prove that pry 2 s-F-limI ðli Þ. Let n be an arbitrary fuzzy semi-open Q-neighbourhood of pry in Y. Then we have pry qn or equivalently r + n(y) > 1. Since r 6 li(y), for every i 2 I, i P i0 we have that li(y) + n(y) > 1, for every i 2 I, i P i0. Thus, liqn, for every i 2 I, i P i0 and therefore pry 2 s-F-limI ðli Þ. The other proofs can be obtained by using the definitions. h Definition 39. A net {li, i 2 I} of fuzzy sets in a fuzzy topological space Y is said to be fuzzy s-convergent to the fuzzy set l if s-F-limI ðli Þ ¼ s-F-limI ðli Þ ¼ l. We then write s-F-limI(li) = l. Theorem 40. Let {li, i 2 I} be a s-convergent net of fuzzy sets in Y. (1) If li1 P li2 for i1 6 i2, then s-F-limI(li) = ^{s-cl(li) : i 2 I}. (2) If li1 6 li2 for i1 6 i2, then s-F-limI(li) = s-cl(_{li : i 2 I}).
Proof. (1) By Theorems 32, 37 and 38, we have ^fs-clðli Þ : i 2 Ig 6 s-F-limI ðs-clðli ÞÞ ¼ s-F-limI ðli Þ 6 s-F-limI ðli Þ ¼ s-F-limI ðs-clðli ÞÞ ¼ ^fs-clðli Þ : i 2 Ig. Thus, s-F-limI(li) = ^{s-cl(li) : i 2 I}.
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(2) By Theorems 32 and 35 we have s-clð_fli : i 2 IgÞ 6 s-F-limI ðli Þ 6 s-F-limI ðli Þ 6 s-clð_fli : i 2 IgÞ. Thus, s-FlimI(li) = s-cl(_{li : i 2 I}). h Theorem 41. Let {li, i 2 I} and {qi, i 2 I} be two s-convergent nets of fuzzy sets in Y. Then the following properties hold: (1) (2) (3) (4)
If li 6 qi for every i 2 I, then s-F-limI(li) 6 s-F-limI(qi), s-F-limI(li _ qi) = s-F-limI(li) _ s-F-limI(qi), s-cl(F-limI (li)) = s-F-limI (li) = s-F-limI(s-cl(li)), If li = l for every i 2 I, then s-F-limI(li) = s-cl(l).
Proof. (1) follows by Theorems 32 and 38. (2) By Theorems 32 and 38 we have s-F-limI ðli _ qi Þ ¼ s-F-limI ðli Þ _ s-F-limI ðqi Þ 6 s-F-limI ðli Þ _ s-F-limI ðqi Þ 6 s-F-limI ðli _ qi Þ 6 s-F-limI ðli _ qi Þ. Thus, s-F-limI(li _ qi) = s-F-limI(li) _ s-F-limI(qi). (3) and (4) follows by Theorems 32 and 38. h Theorem 42. Let n1 and n2 be fuzzy semi-open Q-neighbourhoods of prx and pry in X and Y, respectively. Then the fuzzy set n1 · n2 is a fuzzy semi-open Q-neighbourhood of prðx;yÞ in X · Y. Theorem 43. Let {li, i 2 I} and {qi, i 2 I} be two nets of fuzzy sets in Y. Then the following properties hold: (1) s-F-limI ðli qi Þ 6 s-F-limI ðli Þ s-F-limI ðqi Þ, (2) s-F-limI ðli qi Þ 6 s-F-limI ðli Þ s-F-limI ðqi Þ, (3) If {li, i 2 I} and {qi, i 2 I} are s-convergent nets, then s-F-limI(li · qi) 6 s-F-limI(li) · s-F-limI(qi).
Proof. (1) Let prðx;yÞ 2 s-F-limI ðli qi Þ. We must prove that prðx;yÞ 2 s-F-limI ðli Þ s-F-limI ðqi Þ or equivalently r 6 ðs-F-limðli Þ s-F-limðqi ÞÞðx; yÞ. I
I
Let i0 2 I, n1 be an arbitrary fuzzy semi-open Q-neighbourhood of prx in X and n2 be a constant fuzzy semi-open Qneighbourhood of pry in Y. Then, the fuzzy set n1 · n2 is a fuzzy semi-open Q-neighbourhood of prðx;yÞ in X · Y. Hence, there exists i 2 I, i P i0 such that n1 · n2qli · qi. We have n1qli and n2qqi. Thus, prx 2 s-F-limI ðli Þ. Similarly, we can prove that pry 2 s-F-limI ðqi Þ. Hence prðx;yÞ 2 s-F-limI ðli Þ s-F-limI ðqi Þ. Similarly, we can prove (2) and (3). h Theorem 44. A net {fi, i 2 I} in s-FC(Y, Z) fuzzy s-continuously converges to f 2 s-FC(Y;Z) if and only if s-F-limI ðfi1 ðgÞÞ 6 f 1 ðgÞ for every fuzzy semi-closed subset g of Z. Proof. Let {fi, i 2 I} be a net in s-FC(Y, Z), which fuzzy s-continuously converges to f and let g be an arbitrary fuzzy semi-closed subset of Z. Let p 2 s-F-limI ðfi1 ðgÞÞ and let l be an arbitrary fuzzy semi-open Q-neighbourhood of f(p) in Z. Since the net {fi, i 2 I} fuzzy s-continuously converges to f, there exist a fuzzy semi-open Q-neighbourhood f of p in Y and an element i0 2 I such that fi(f) 6 l, for every i 2 I, i P i0 by Theorem 8. On the other hand, there exists an element i P i0 such that fqf 1 i ðgÞ. Hence, fi(f)qg and therefore lqg . This means that f(p) 2 s-cl(g) = g. Thus, p 2 f1(g). Conversely, let {fi, i 2 I} be a net in s-FC(Y, Z) and f 2 s-FC(Y, Z) such that s-F-limI ðfi1 ðgÞÞ 6 f 1 ðgÞ for every fuzzy semi-closed subset g of Z. We prove that the net {fi, i 2 I} fuzzy s-continuously converges to f. Let p be a fuzzy point of Y and l be a fuzzy semi-open Q-neighbourhood of f(p) in Z. Since p62f1(g), where g = lc we have p 62 s-F-limI ðfi1 ðgÞÞ. This means that there exists an element i0 2 I and a fuzzy semi-open Q-neighbourhood f of p in Y such that fi1 ðgÞqf, for every i 2 I, i P i0. Then we have f 6 ðfi1 ðgÞÞc ¼ fi1 ðgc Þ ¼ fi1 ðlÞ and, therefore, fi(f) 6 l, for every i 2 I, i P i0, that is the net {fi, i 2 I} fuzzy s-continuously converges to f.
h
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On fuzzy upper and lower s-limit sets Erdal Ekici Department of Mathematics, Canakkale Onsekiz Mart University, Terzioglu Campus, 17020 Canakkale, Turkey Accepted 10 August 2005
Abstract The aim of this paper is to introduce the notions of fuzzy upper and lower s-limits sets. Properties and basic relationships among fuzzy upper s-limit set, fuzzy lower s-limit set, fuzzy semi-continuity are investigated. Possible applications to quantum physics and computer graphics are touched upon. We speculate that s-convergence and s-properties may be relavant to the physics of fractal and cantorian spacetime. 2005 Elsevier Ltd. All rights reserved.
1. Introduction This paper will give some new topological properties (for example s-limit sets, s-convergence, s-continuously convergence, s-separation properties) which have been found to be very useful in the study of certain objects of digital topology. Thus we may stress once more the importance of s-convergence as a branch of them and the possible applications in quantum physics [4–7] and computer graphics [8–10]. We speculate that s-convergence and s-properties may be relavant to the physics of fractal and cantorian spacetime [4,6,7,13]. In this paper, the symbol I will denote the unit interval [0, 1]. A fuzzy set in X is a function with domain X and values in I, that is, an element of IX. Let l 2 IX. The subset of X in which l assumes nonzero values, is known as the support of l [18]. A member l of IX is contained in a member q of IY denoted l 6 q if and only if l(x) 6 q(x), for every x 2 X [18]. By l · q we denote the fuzzy set in X · Y for which (l · q)(x, y) = min{l(x),q (y)}, for every (x, y) 2 X · Y. For l 2 IX, the fuzzy set lc is defined by lc(x) = 1 l(x) for every x 2 X. A fuzzy set in X is called a fuzzy point if and only if it takes the value 0 for all y 2 X except one, say, x 2 X. If its value at x is e (0 < e 6 1) we denote the fuzzy point by pex , where the point x is called its support. The class of all fuzzy points in X is denoted by v [11,14]. The fuzzy point pex is said to be contained in a fuzzy set l or to belong to l, denoted by pex 2 l, if and only if e 6 l(x). Chang [2] introduced the concept of fuzzy topology. A fuzzy set l in a fuzzy topological space (X, s) is called a neighbourhood of a fuzzy point pex if and only if there exists a f 2 s such that pex 2 f 6 l [11]. A fuzzy point pex is said to be quasi-coincident with l denoted by pex ql if and only if e > lc(x) or e + l(x) > 1 [11]. A fuzzy set l is said to be quasi-coincident with q, denoted lqq, if and only if there exists x 2 X such that l(x) > qc(x) or l(x) + q(x) > 1 [11]. If l does not quasi-coincident with q, then we write l qq. A fuzzy set l of a fuzzy topological space (X, s) is called a fuzzy semi-open set if l 6 cl(int(l)) [1]. Also, a fuzzy set l of a fuzzy topological space (X, s) is called fuzzy semi-closed if the fuzzy set lc is fuzzy semi-open. Finally, by s-cl(l)
E-mail address: [email protected] 0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.033
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and s-int(l) we denote the fuzzy set: s-cl(l) = ^{q 2 IX : q is fuzzy semi-closed, l 6 q}, s-int(l) = _{q 2 IX : q is fuzzy semi-open, q 6 l}, respectively [17]. The family of all fuzzy semi-open sets of X is denoted by FSO(X). A fuzzy set l in a fuzzy topological space (X, s) is called a Q-neighbourhood of pex if and only if there exists q 2 s such that pex qq and q 6 l. In what follows by Nðpex Þ we denote the family of all fuzzy open Q-neighbourhoods of the fuzzy point pex in X. A fuzzy point pex 2 clðlÞ if and only if each Q-neighbourhood of pex is quasi-coincident with l [11]. A fuzzy set l in a fuzzy space X is called a fuzzy semi-neighborhood of a fuzzy point pex if there exists a g 2 FSO(X) such that pex 2 g 6 l. A fuzzy set l in a fuzzy space X is called a fuzzy Q-s-neighborhood of pex if there exists q 2 FSO(X) such that pex qq and q 6 l. The set N s ðpex Þ with the relation 6* (that is, n1 6 *n2 if and only if n2 6 n1) form a directed set. The basic knowledges related with the fuzzy functions can be seen in [2,3,12,15,16].
2. Fuzzy s-continuously converge Definition 1. A function f from a fuzzy topological space Y into a fuzzy topological space Z is called fuzzy semicontinuous if and only if for every fuzzy point pex in Y and every fuzzy semi-open Q-neighbourhood f of f ðpex Þ, there exists a fuzzy semi-open Q-neighbourhood n of pex such that f(n) 6 f. Let Y and Z be fuzzy topological spaces. By s-FC(Y, Z) we denote the set of all fuzzy semi-continuous functions of Y into Z. Definition 2. Let (X, s) be a fuzzy topological space and let {pi, i 2 I} be a net of fuzzy points in X. We say that the fuzzy net {pi, i 2 I} fuzzy s-converges to a fuzzy point p of X if for every fuzzy semi-open Q-neighbourhood n of p in X there exists i0 2 I such that piqn, for every i 2 I and i P i0. Theorem 3. Let l be a fuzzy set of a fuzzy space X. Then, a fuzzy point pex 2 s-clðlÞ if and only if for every n 2 FSO(X) for which pex qn we have nql. Proof. The fuzzy point pex 2 s-clðlÞ if and only if pex 2 f, for every fuzzy semi-closed set f of X for which l 6 f. Equivalently pex 2 s-clðlÞ if and only if e 6 1 n(x), for every fuzzy semi-open set n for which l 6 1 n. Thus pex 2 s-clðlÞ if and only if n(x) 6 1 e, for every fuzzy semi-open set n for which n 6 1 l. So, pex 2 s-clðlÞ if and only if for every fuzzy semi-open set n of X such that n(x) > 1 e we have n i X l. Therefore, pex 2 s-clðlÞ if and only if for every fuzzy semi-open set n of X such that n(x) + e > 1 we have nql. Thus, pex 2 s-clðlÞ if and only if for every fuzzy semi-open set n of X such that pex qn we have nql. h Theorem 4. Let f : Y ! Z be a fuzzy semi-continuous map, p be a fuzzy point in Y and n, f be fuzzy semi-open Q-neighqf. bourhoods of p and f(p), respectively such that f(n) i f. Then there exists a fuzzy point p1 in Y such that p1qn and f ðp1 Þ Proof. Since f(n) i f, we have n i f1(f). Thus there exists x 2 Y such that n(x) > f1(f)(x) or n(x) f1(f)(x) > 0 and therefore n(x) + 1 f1(f)(x) > 1 or n(x) + (f1(f))c(x) > 1. Let (f1(f))c(x) = r. Clearly, for the fuzzy point prx we have prx qn and prx 2 ðf 1 ðfÞÞc . Hence for the fuzzy point p1 ¼ prx we have p1qn and f ðp1 Þ qf. h Theorem 5. A map f from a fuzzy topological space Y into a fuzzy topological space Z is fuzzy semi-continuous if and only if for every fuzzy point p of Y and for every net {pi, i 2 I} which fuzzy s-converges to p the net {f(pi), i 2 I} of Z fuzzy s-converges to f(p). Definition 6. A net {fj, j 2 J} in s-FC(Y, Z) fuzzy s-continuously converges to f 2 s-FC(Y, Z) if and only if for every fuzzy net {pi, i 2 I} in Y which fuzzy s-converges to a fuzzy point p in Y we have that the fuzzy net {fj(pi),(i, j) 2 I · J} fuzzy s-converges to the fuzzy point f(p) in Z. Theorem 7. The following properties hold: (1) If {fi, i 2 I} is a net in s-FC(Y, Z) such that fi = f, for every i 2 I, then the {fi, i 2 I} fuzzy s-continuously converges to f 2 s-FC(Y, Z). (2) If {fi, i 2 I} is a net in s-FC(Y, Z), which fuzzy s-continuously converges to f 2 s-FC(Y, Z) and {gj, j 2 J} be a subnet of {fi, i 2 I}, then the net {gj, j 2 J} fuzzy s-continuously converges to f. (3) If {fi, i 2 I} is a net in s-FC(Y, Z) which does not fuzzy s-continuously converges to f 2 s-FC(Y, Z), then there exists no subnet of {fi, i 2 I}, which fuzzy continuously converges to f.
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Theorem 8. A net {fj, j 2 J} in s-FC(Y, Z) fuzzy s-continuously converges to f 2 s-FC(Y, Z) if and only if for every fuzzy point p in Y and for every fuzzy semi-open Q-neighbourhood f of f(p) in Z there exist an element j0 2 J and a fuzzy semiopen Q-neighbourhood n of p in Y such that fj(n) 6 f, for every j P j0, j 2 J. Proof. Let p be a fuzzy point in Y and let f be a fuzzy semi-open Q-neighbourhood of f(p) in Z such that for every j 2 J and for every fuzzy semi-open Q-neighbourhood n of p in Y there exists j 0 P j such that fj0 ðnÞif. Then for every fuzzy semi-open Q-neighbourhood n of p in Y we can choose a fuzzy point pn in Y by Theorem 4 such that pnqn and fj0 ðpn Þ qf. Clearly, the fuzzy net {pn,n 2 N(p)} fuzzy s-converges to p, but the fuzzy net ffj ðpn Þ; ðn; jÞ 2 NðpÞ J g does not fuzzy s-converge to f(p) in Z. Conversely, let {pi, i 2 I} be a fuzzy net in s-FC(Y, Z) which s-converges to the fuzzy point p in Y and let f be an arbitrary fuzzy semi-open Q-neighbourhood of f(p) in Z. By assumption there exists a fuzzy semi-open Qneighbourhood n of p in Y and an element j0 2 J such that fj(n) 6 f, for every j P j0, j 2 J. Since the fuzzy net {pi, i 2 I} s-converges to p in Y. There exists i0 2 I such that piqn, for every i 2 I, i P i0. Let (i0, j0) 2 I · J. Then for every (i, j) 2 I · J, (i, j) P (i0, j0) we have fj(pi)qfj(n) and fj(n) 6 f, that is fj(pi)qf. Thus the net {fj(pi), (i, j) 2 I · J} fuzzy sconverges to f(p) and the net {fj, j 2 J} fuzzy s-continuously converges to f. h Definition 9. A fuzzy set l of a fuzzy space X is called semi-generalized closed (briefly f-sg-closed) if s-cl(l) 6 n whenever l 6 n and n fuzzy semi-open set of X. Definition 10. A fuzzy space X is called s-T1 if every fuzzy point is fuzzy semi-closed. Theorem 11. A fuzzy space X is s-T1 if and only if for each x 2 X and each e 2 [0, 1] there exists a fuzzy semi-open set l such that l(x) = 1 e and l(y) = 1 for y 5 x. Proof. ()) Let e = 0. We set l = X. Then l is fuzzy semi-open set such that l(x) = 1 0 and l(y) = 1 for y 5 x. Now, let e 2 (0, 1] and x 2 X. We set l ¼ ðpex Þc . The set l is fuzzy semi-open such that l(x) = 1 e and l(y) = 1 for y 5 x. (() Let pex be an arbitrary fuzzy point of X. We prove that the fuzzy point pex is fuzzy semi-closed. By assumption there exists a fuzzy semi-open set l such that l(x) = 1 e and l(y) = 1 for y 5 x. Clearly, lc ¼ pex . Thus the fuzzy point pex is fuzzy semi-closed and therefore the fuzzy space X is s-T1. h Definition 12. A fuzzy space X is called a quasi-s-T1 if for any fuzzy points pex and pay for which suppðpex Þ ¼ x 6¼ suppðpay Þ ¼ y, there exists a fuzzy semi-open set n such that pex 2 n and pay 62 n and another g such that pex 62 g and pay 2 g. 0;75 Example 13. Let (X, s) be a fuzzy topological space such that X = {x, y} and s ¼ f£; X ; p0;30 x ; p x g. The fuzzy topological space X is quasi-s-T1 but it is not s-T1.
Definition 14. A fuzzy space X is called a s-T2 space if for any fuzzy points pex and pay for which suppðpex Þ ¼ x 6¼ suppðpay Þ ¼ y, there exist two fuzzy semi-open Q-s-neighbourhoods n and g of pex and pay , respectively, such that n ^ g = B. Definition 15. A fuzzy point pex is called weak (respectively, strong) if e 6 12 (respectively, e > 12) [11]. Theorem 16. If X is a quasi-s-T1 fuzzy space and pex a weak fuzzy point in X, then ðpex Þc is a fuzzy semi-neighborhood of each fuzzy point pay with y 5 x. Proof. Let y 5 x and pay be a fuzzy point of X. Since the space X is a quasi-s-T1 there exists a fuzzy semi-open n of X such that pay 2 n and pex 62 n. This implies that e > n(x). Also, e 6 12. Thus n(x) 6 1 e. Therefore nðyÞ 6 1 ¼ ðpex Þc ðyÞ; for every y 2 Xn{x}. So n 6 ðpex Þc . Hence, the fuzzy point pex is a semi-neighborhood of pay .
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Definition 17. A fuzzy space X is called a s-regular space if for any fuzzy point pex and a fuzzy semi-closed set f not containing pex , there exist n, g 2 FSO(X) such that pex 2 n, f 6 g and n ^ g = B. Theorem 18. If X is a s-regular fuzzy space, then for any strong fuzzy point pex and any fuzzy semi-open set n containing pex , there exists a fuzzy semi-open set q containing pex such that s-cl(q) 6 n. Proof. Suppose that pex is any strong fuzzy point contained in n 2 FSO(X). Then 12 < e 6 nðxÞ. Thus the complement of n, that is the fuzzy set nc, is a fuzzy semi-closed set to which does not belong the fuzzy point pex . Thus, there exist q, g 2 FSO(X) such that pex 2 q and nc 6 g with q ^ g = B. Hence, we have q 6 gc and s-clðqÞ 6 s-clðgc Þ ¼ gc . Now nc 6 g implies gc 6 n. This means that s-cl(q) 6 n which complete the proof.
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Theorem 19. If X is a fuzzy s-regular space, then the strong fuzzy points in X are f-sg-closed. Proof. Let pex be any strong fuzzy point in X and n be a fuzzy semi-open set such that pex 2 n. By the previous theorem there exists a q 2 FSO(X) such that pex 2 q and s-cl(q) 6 n. We have s-clðpex Þ 6 s-clðqÞ 6 n. Hence, the fuzzy point pex is f-sg-closed. h Definition 20. A fuzzy space X is called a weakly s-regular space if for any weak fuzzy point pex and a fuzzy semi-closed set f not containing pex , there exist n, g 2 FSO(X) such that pex 2 n, f 6 g and n ^ g = B. Definition 21. A fuzzy set n in a fuzzy space X is said to be fuzzy s-nearly crisp if s-cl(n) ^ (s-cl(n))c = B. Theorem 22. Let X be a fuzzy space. If for any weak fuzzy point pex and any n 2 FSO(X) containing pex , there exists a fuzzy semi-open and s-nearly crisp fuzzy set q containing pex such that s-cl(q) 6 n, then X is fuzzy weakly s-regular. Proof. Assume that f is a fuzzy semi-closed set not containing the weak fuzzy point pex . Then fc is a fuzzy semi-open set containing pex . By hypothesis, there exists a fuzzy semi-open and s-nearly crisp fuzzy set q such that pex 2 q and s-cl(q) 6 fc. We set g = s-int(s-cl(q)) and l = 1 s-cl(q). Then g is fuzzy semi-open, pex 2 g and f 6 l. We are going to prove that l ^ g = B. Now assume that there exists y 2 X such that (g ^ l)(y) = a 5 B. Then pay 2 g ^ l. Hence, pay 2 s-clðqÞ and pay 2 ðs-clðqÞÞc . This is a contradiction since q is s-nearly crisp. Hence, the fuzzy space X is weakly s-regular. h Definition 23. Let l be a fuzzy set of a fuzzy space X. A fuzzy point pex is called a semi-boundary point of a fuzzy set l if and only if pex 2 s-clðlÞ ^ ðX s-clðlÞÞ. By s-bd(l) we denote the fuzzy set s-cl(l) ^ (X s-cl(l)). Theorem 24. Let X be a fuzzy space. Suppose that pex and pay be weak and strong fuzzy points, respectively. If pex is semigeneralized closed, then pay 2 s-clðpex Þ ) pex 2 s-clðpay Þ. Proof. Suppose that pay 2 s-clðpex Þ and pex 62 s-clðpay Þ. Then s-clðpay ÞðxÞ < e. Also e 6 12. Thus s-clðpay ÞðxÞ 6 1 e and therefore e 6 1 s-clðpay ÞðxÞ. So pex 2 ðs-clðpay ÞÞc . But pex is semi-generalized closed and ðs-clðpay ÞÞc fuzzy semi-open. Hence, s-clðpex Þ 6 ðs-clðpay ÞÞc . By assumption we have pay 2 s-clðpex Þ. Thus, pay 2 ðs-clðpay ÞÞc . We prove that this is a contradiction. Indeed, we have a 6 1 s-clðpay ÞðyÞ or s-clðpay ÞðyÞ 6 1 a. Also pay 2 s-clðpay Þ. Thus a 6 1 a. But pay is a strong fuzzy point, that is a > 12. So the above relation a 6 1 a is a contradiction. Hence, pex 2 s-clðpay Þ. h Theorem 25. Let l be a fuzzy set of a fuzzy space X. Then l _ s-bd(l) 6 s-cl(l}. Proof. Let pex 2 l _ s-bdðlÞ. Then pex 2 l or pex 2 s-bdðlÞ. If pex 2 s-bdðlÞ, then pex 2 s-clðlÞ. Let us suppose that pex 2 l. We have s-cl(l) = ^{f : f 2 IX, f is semi-closed and l 6 f}. So, if pex 2 l, then pex 2 f, for every fuzzy semi-closed f of X for which l 6 f and therefore pex 2 s-clðlÞ. h
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Definition 26. A fuzzy point pex in a fuzzy space X is said to be (1) well-semi-closed if there exists pay 2 s-clðpex Þ such that suppðpex Þ 6¼ suppðpay Þ, (2) just-semi-closed if the fuzzy set s-clðpex Þ is again fuzzy point. Clearly, in a fuzzy s-T1 space every fuzzy point is just-semi-closed. Theorem 27. If X is a fuzzy space and pex is a f-sg-closed but and well semi-closed fuzzy point, then X is not quasi-s-T1 space. Proof. Let X be a fuzzy quasi-s-T1 space. By the fact pex is well-semi-closed, there exists a fuzzy point pay with suppðpex Þ 6¼ suppðpay Þ such that pay 2 s-clðpex Þ. Then there exists an n 2 FSO(X) such that pex 2 n and pay 62 n. Therefore s-clðpex Þ 6 n and pay 2 n. But this is a contradiction and hence X cannot be quasi-s-T1 space. h Theorem 28. Let X be a fuzzy space. If pex and pax are two fuzzy points such that e < a and pax is fuzzy semi-open, then pex is just-semi-closed if it is f-sg-closed. Proof. We prove that the fuzzy set s-clðpex Þ is again a fuzzy point. We have pex 2 pax and the fuzzy set pax is fuzzy semiopen. Since pex is f-sg-closed we have s-clðpex Þ 6 pax . Thus s-clðpex ÞðxÞ 6 a and s-clðpex ÞðzÞ 6 0, for every z 2 Xn{x}. So the fuzzy set s-clðpex Þ is a fuzzy point. h 3. Fuzzy upper and lower s-limit sets Definition 29. Let I be a directed set. Let X be an ordinary set. Let v be the collection of all fuzzy points in X. The function S : I ! v is called a fuzzy net in X. For every i 2 I, S(I) is often denoted by si and hence a net S is often denoted by {si, i 2 I} [11]. Definition 30. Let S = {si, i 2 I} be a fuzzy net in X. S is said to be quasi-coincident with l if for each i 2 I, si is quasicoincident with l. A fuzzy net {gj, j 2 J} in X is called a fuzzy subnet of a fuzzy net {si, i 2 I} in X if there is a map N:J ! I such that (a) gj = sN(j) and (b) for the element i0 2 I there is j0 2 J such that if j P j0, j 2 J, then N(j) P i0. Definition 31. Let {li, i 2 I} be a net of fuzzy sets in a fuzzy topological space Y. Then s-F-limI ðli Þ, we denote fuzzy upper s-limit of the net {li, i 2 I} in IY, that is, the fuzzy set which is the union of all fuzzy points prx in Y such that for every i0 2 I and for every fuzzy semi-open Q-neighbourhood n of prx in Y there exists an element i 2 I for which i P i0 and liqn. In other cases we set s-F-limI ðli Þ ¼ £. Theorem 32. Let {li, i 2 I} and {qi, i 2 I} be two nets of fuzzy sets in Y. Then the following properties hold: (1) (2) (3) (4) (5) (6) (7) (8)
The fuzzy upper s-limit is semi-closed, s-F-limI ðli Þ ¼ s-F-limI ðs-clðli ÞÞ, If li = l for every i 2 I, then s-F-limI ðli Þ ¼ s-clðlÞ, The fuzzy upper s-limit is not affected by changing a finite number of the li, s-F-limI ðli Þ 6 s-clð_fli : i 2 IgÞ, If li 6 qi for every i2I, then s-F-limI ðli Þ 6 s-F-limI ðqi Þ, s-F-limI ðli _ qi Þ ¼ s-F-limI ðli Þ _ s-F-limI ðqi Þ, s-F-limI ðli ^ qi Þ 6 s-F-limI ðli Þ ^ s-F-limI ðqi Þ.
Proof. (1) It is sufficient to prove that s-clðs-F-limI ðli ÞÞ 6 s-F-limI ðli Þ. Let pry 2 s-clðs-F-limI ðli ÞÞ and let n be an arbitrary fuzzy semi-open Q-neighbourhood of pry . Then, we have nqs-F-limI ðli Þ. Hence, there exists an element y 0 2 Y such that nðy 0 Þ þ s-F-limI ðli Þðy 0 Þ > 1. Let s-F-limI ðli Þðy 0 Þ ¼ a. Then, for the fuzzy point pay 0 in Y we have pay 0 qn and pay 0 2 s-F-limI ðli Þ. Thus, for every element i0 2 I there exists i P i0, i 2 I such that liqn. This means that pry 2 s-F-limI ðli Þ. (2) Clearly, it is sufficient to prove that for every fuzzy semi-open set n the condition nqli is equivalent to nqs-cl(li). Let nqli. Then there exists an element y 2 Y such that n(y) + li(y) > 1. Since li 6 s-cl(li) we have n(y) + s-cl(li)(y) > 1 and therefore n qs-cl(li).
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Conversely, let nqs-cl(li). Then there exists an element y 2 Y such that n(y) + s-cl(li)(y) > 1. Let s-cl(li)(y) = r. Then pry 2 s-clðli Þ and the fuzzy semi-open set n is a fuzzy semi-open Q-neighbourhood of pry . Thus, nqli. (3), (4): Obvious. (5) Let pry 2 s-F-limI ðli Þ and let n be a fuzzy semi-open Q-neighbourhood of pry in Y. Then for every i0 2 I there exists i 2 I, i P i0 such that liqn and therefore _{li : i 2 I} qn. Thus, pry 2 s-clð_fli : i 2 IgÞ. (7) Clearly, li 6 li _ qi and qi 6 li _ qi, for every i 2 I. Hence, by (6), s-F-limðli Þ 6 s-F-limðli _ qi Þ I
I
and s-F-limðqi Þ 6 s-F-limðli _ qi Þ. I
I
Thus, s-F-limI ðli Þ _ s-F-limI ðqi Þ 6 s-F-limI ðli _ qi Þ. Conversely, let pry 2 s-F-limI ðli _ qi Þ. We prove that pry 2 s-F-limI ðli Þ _ s-F-limI ðqi Þ. Let us suppose that pry 62 s-F-limI ðli Þ _ s-F-limI ðqi Þ. Then pry 62 s-F-limI ðli Þ and pry 62 s-F-limI ðqi Þ. Hence, there exists a fuzzy semi-open qn1 , for every i 2 I, i P i1. Also, there exists a fuzzy semiQ-neighbourhood n1 of pry and an element i1 2 I such that li qn2 , for every i 2 I, i P i2. Let n = n1 _ n2 and let i0 2 I open Q-neighbourhood n2 of pry and an element i2 2 I such that qi such that i0 P i1 and i0 P i2. Then the fuzzy set n is a fuzzy semi-open Q-neighbourhood of pry and li _ qi qn, for every i 2 I, i P i0. Since pry 2 s-F-limI ðli _ qi Þ this is a contradiction. Thus, pry 2 s-F-limI ðli Þ _ s-F-limI ðqi Þ. The other proofs can be obtained by using the definitions.
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Definition 33. Let {li, i 2 I} be a net of fuzzy sets in a fuzzy topological space Y. Then by s-F-limI ðli Þ, we denote the fuzzy lower s-limit of the net {li, i 2 I} in IY, that is, the fuzzy set which is the union of all fuzzy points prx in Y such that for every fuzzy semi-open Q-neighbourhood n of prx in Y there exists an element i0 2 I such that liqn, for every i 2 I, i P i0. In other case we set s-F-limI ðli Þ ¼ £. Theorem 34. For the fuzzy upper and lower s-limit we have s-F-limI ðli Þ 6 s-F-limI ðli Þ. Proof. The proof of this theorem follows by the Definitions 31 and 33.
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Theorem 35. Let {li, i 2 I} be a net of fuzzy sets in Y such that li1 6 li2 if and only if i1 6 i2. Then s-clð_fli : i 2 IgÞ ¼ s-F-limI ðli Þ. Proof. Let pry 2 s-clð_fli : i 2 IgÞ and let n be a fuzzy semi-open Q-neighbourhood of pry in Y. Then nq _ {li : i 2 I}. Hence, there exists i0 2 I such that nqli0 . By assumption we have nqli, for every i 2 I, i P i0. Thus pry 2 s-F-limI ðli Þ. Conversely, let pry 2 s-F-limI ðli Þ and let n be an arbitrary fuzzy semi-open Q-neighbourhood of pry in Y. Then there exists an element i0 2 I such that nqli, for every i 2 I, i P i0. Hence nq _ {li : i 2 I} and therefore pry 2 s-clð_fli : i 2 IgÞ. h Theorem 36. Let {li, i 2 I} be a net of fuzzy sets in Y. Then we have s-F -limI ðli Þ ¼ ^fs-clð_fli : i P i0 gÞ : i0 2 Ig. Proof. Let pry 2 s-F-limI ðli Þ and let i0 2 I. We prove that pry 2 s-clð_fli : i P i0 gÞ. Let n be an arbitrary fuzzy semi-open Q-neighbourhood of pry in Y. Then, there exists i P i0, i 2 I such that nqli. Thus, nq _ {li : i P i0} and therefore pry 2 s-clð_fli : i P i0 gÞ. Conversely, let pry 2 ^fs-clð_fli : i P i0 gÞ : i0 2 Ig. Then, we have pry 2 s-clð_fli : i P i0 gÞ, for every i0 2 I. We prove that pry 2 s-F-limI ðli Þ. Let n be an arbitrary fuzzy semi-open Q-neighbourhood of pry in Y and let i0 2 I. Then nq _ {li : i P i0}. We prove that there exists i 2 I, i P i0 such that liqn. Let us suppose that nqli ,for every i 2 I, i P i0. Then, for every i 2 I, i P i0 and for every y 2 Y we have n(y) + li(y) 6 1 and therefore
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nðyÞ þ ð_fli : i P i0 gÞðyÞ 6 1; which is a contradiction. Hence, pry 2 s-F-limI ðli Þ.
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Theorem 37. Let {li, i 2 I} be a net of fuzzy semi-closed sets in Y such that li1 6 li2 if and only if i2 6 i1. Then s-F-limI ðli Þ ¼ ^fli : i 2 Ig. Proof. Let pry 2 ^fli : i 2 Ig. Then pry 2 li or r 6 li(y) for every i 2 I. Let i0 2 I and n be a fuzzy semi-open Q-neighbourhood of pry , that is r + n(y) > 1. Then there exists i 2 I, i P i0 such that li(y) + n(y) P r + n(y) > 1. Hence, liqn and therefore pry 2 s-F-limI ðli Þ. Conversely, let pry 2 s-F-limI ðli Þ and let pry 62 ^fli : i 2 Ig. Then there exists i0 2 I such that pry 62 li0 , that is r > li0 ðyÞ. Let n ¼ ðli0 Þc . Then n is fuzzy semi-open Q-neighbourhood of pry and for every i P i0, n qli , which is a contradiction. h Theorem 38. Let {li, i 2 I} and {qi, i 2 I} be two nets of fuzzy sets in Y. Then the following properties hold: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
If li 6 qi for every i 2 I, then s-F-limI ðli Þ 6 s-F-limI ðqi Þ, s-F-limI ðli _ qi Þ P s-F-limI ðli Þ _ s-F-limI ðqi Þ, s-F-limI ðli ^ qi Þ 6 s-F-limI ðli Þ ^ s-F-limI ðqi Þ, The fuzzy lower s-limit is semi-closed, s-F-limI ðli Þ ¼ s-F-limI ðs clðli ÞÞ, If li = l for every i 2 I, then s-F-limI ðli Þ ¼ s-clðlÞ, The fuzzy lower s-limit is not affected by changing a finite number of the li, ^fli : i 2 Ig 6 s-F-limI ðli Þ, s-F-limI ðli Þ 6 s-clð_fli : i 2 IgÞ, _f^fli : i P i0 g : i0 2 Ig 6 s-F-limI ðli Þ.
Proof. (2) Let pry 2 s-F-limI ðli Þ _ s-F-limI ðqi Þ. Then either pry 2 s-F-limI ðli Þ or pry 2 s-F-limI ðqi Þ. Let pry 2 s-F-limI ðli Þ. Then for every fuzzy semi-open Q-neighbourhood n of pry in Y there exists an element i0 2 I such that liqn, for every i P i0, i 2 I. Also li 6 li _ qi. Thus (li _ qi)qn, for every i 2 I, i P i0 and therefore pry 2 s-F-limI ðli _ qi Þ. The other proofs can be obtained by using the definitions. (8) Let pry 2 ^fli : i 2 Ig. We prove that pry 2 s-F-limI ðli Þ. Let us suppose that pry 62 s-F-limI ðli Þ. Then there exists a qn. This means that fuzzy semi-open Q-neighbourhood n of pry such that for every i 2 I there exists i 0 P i for which li0 li0 ðxÞ þ nðxÞ 6 1, for every x 2 Y. Now, since pry 2 ^fli : i 2 Ig and n is a fuzzy semi-open Q-neighbourhood of pry we have that r 6 li(y), for every i 2 I and r + n(y) > 1. Thus, li(y) + n(y) > 1, for every i 2 I. By the above this is a contradiction. Hence, pry 2 s-F-limI ðli Þ. (10) Let pry 2 _f^fli : i P i0 g : i0 2 Ig. Then, there exists i0 2 I such that pry 2 ^fli : i P i0 g. Hence, pry 2 li , for every i 2 I, i P i0 and therefore r 6 li(y), for every i 2 I, i P i0. We prove that pry 2 s-F-limI ðli Þ. Let n be an arbitrary fuzzy semi-open Q-neighbourhood of pry in Y. Then we have pry qn or equivalently r + n(y) > 1. Since r 6 li(y), for every i 2 I, i P i0 we have that li(y) + n(y) > 1, for every i 2 I, i P i0. Thus, liqn, for every i 2 I, i P i0 and therefore pry 2 s-F-limI ðli Þ. The other proofs can be obtained by using the definitions. h Definition 39. A net {li, i 2 I} of fuzzy sets in a fuzzy topological space Y is said to be fuzzy s-convergent to the fuzzy set l if s-F-limI ðli Þ ¼ s-F-limI ðli Þ ¼ l. We then write s-F-limI(li) = l. Theorem 40. Let {li, i 2 I} be a s-convergent net of fuzzy sets in Y. (1) If li1 P li2 for i1 6 i2, then s-F-limI(li) = ^{s-cl(li) : i 2 I}. (2) If li1 6 li2 for i1 6 i2, then s-F-limI(li) = s-cl(_{li : i 2 I}).
Proof. (1) By Theorems 32, 37 and 38, we have ^fs-clðli Þ : i 2 Ig 6 s-F-limI ðs-clðli ÞÞ ¼ s-F-limI ðli Þ 6 s-F-limI ðli Þ ¼ s-F-limI ðs-clðli ÞÞ ¼ ^fs-clðli Þ : i 2 Ig. Thus, s-F-limI(li) = ^{s-cl(li) : i 2 I}.
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(2) By Theorems 32 and 35 we have s-clð_fli : i 2 IgÞ 6 s-F-limI ðli Þ 6 s-F-limI ðli Þ 6 s-clð_fli : i 2 IgÞ. Thus, s-FlimI(li) = s-cl(_{li : i 2 I}). h Theorem 41. Let {li, i 2 I} and {qi, i 2 I} be two s-convergent nets of fuzzy sets in Y. Then the following properties hold: (1) (2) (3) (4)
If li 6 qi for every i 2 I, then s-F-limI(li) 6 s-F-limI(qi), s-F-limI(li _ qi) = s-F-limI(li) _ s-F-limI(qi), s-cl(F-limI (li)) = s-F-limI (li) = s-F-limI(s-cl(li)), If li = l for every i 2 I, then s-F-limI(li) = s-cl(l).
Proof. (1) follows by Theorems 32 and 38. (2) By Theorems 32 and 38 we have s-F-limI ðli _ qi Þ ¼ s-F-limI ðli Þ _ s-F-limI ðqi Þ 6 s-F-limI ðli Þ _ s-F-limI ðqi Þ 6 s-F-limI ðli _ qi Þ 6 s-F-limI ðli _ qi Þ. Thus, s-F-limI(li _ qi) = s-F-limI(li) _ s-F-limI(qi). (3) and (4) follows by Theorems 32 and 38. h Theorem 42. Let n1 and n2 be fuzzy semi-open Q-neighbourhoods of prx and pry in X and Y, respectively. Then the fuzzy set n1 · n2 is a fuzzy semi-open Q-neighbourhood of prðx;yÞ in X · Y. Theorem 43. Let {li, i 2 I} and {qi, i 2 I} be two nets of fuzzy sets in Y. Then the following properties hold: (1) s-F-limI ðli qi Þ 6 s-F-limI ðli Þ s-F-limI ðqi Þ, (2) s-F-limI ðli qi Þ 6 s-F-limI ðli Þ s-F-limI ðqi Þ, (3) If {li, i 2 I} and {qi, i 2 I} are s-convergent nets, then s-F-limI(li · qi) 6 s-F-limI(li) · s-F-limI(qi).
Proof. (1) Let prðx;yÞ 2 s-F-limI ðli qi Þ. We must prove that prðx;yÞ 2 s-F-limI ðli Þ s-F-limI ðqi Þ or equivalently r 6 ðs-F-limðli Þ s-F-limðqi ÞÞðx; yÞ. I
I
Let i0 2 I, n1 be an arbitrary fuzzy semi-open Q-neighbourhood of prx in X and n2 be a constant fuzzy semi-open Qneighbourhood of pry in Y. Then, the fuzzy set n1 · n2 is a fuzzy semi-open Q-neighbourhood of prðx;yÞ in X · Y. Hence, there exists i 2 I, i P i0 such that n1 · n2qli · qi. We have n1qli and n2qqi. Thus, prx 2 s-F-limI ðli Þ. Similarly, we can prove that pry 2 s-F-limI ðqi Þ. Hence prðx;yÞ 2 s-F-limI ðli Þ s-F-limI ðqi Þ. Similarly, we can prove (2) and (3). h Theorem 44. A net {fi, i 2 I} in s-FC(Y, Z) fuzzy s-continuously converges to f 2 s-FC(Y;Z) if and only if s-F-limI ðfi1 ðgÞÞ 6 f 1 ðgÞ for every fuzzy semi-closed subset g of Z. Proof. Let {fi, i 2 I} be a net in s-FC(Y, Z), which fuzzy s-continuously converges to f and let g be an arbitrary fuzzy semi-closed subset of Z. Let p 2 s-F-limI ðfi1 ðgÞÞ and let l be an arbitrary fuzzy semi-open Q-neighbourhood of f(p) in Z. Since the net {fi, i 2 I} fuzzy s-continuously converges to f, there exist a fuzzy semi-open Q-neighbourhood f of p in Y and an element i0 2 I such that fi(f) 6 l, for every i 2 I, i P i0 by Theorem 8. On the other hand, there exists an element i P i0 such that fqf 1 i ðgÞ. Hence, fi(f)qg and therefore lqg . This means that f(p) 2 s-cl(g) = g. Thus, p 2 f1(g). Conversely, let {fi, i 2 I} be a net in s-FC(Y, Z) and f 2 s-FC(Y, Z) such that s-F-limI ðfi1 ðgÞÞ 6 f 1 ðgÞ for every fuzzy semi-closed subset g of Z. We prove that the net {fi, i 2 I} fuzzy s-continuously converges to f. Let p be a fuzzy point of Y and l be a fuzzy semi-open Q-neighbourhood of f(p) in Z. Since p62f1(g), where g = lc we have p 62 s-F-limI ðfi1 ðgÞÞ. This means that there exists an element i0 2 I and a fuzzy semi-open Q-neighbourhood f of p in Y such that fi1 ðgÞqf, for every i 2 I, i P i0. Then we have f 6 ðfi1 ðgÞÞc ¼ fi1 ðgc Þ ¼ fi1 ðlÞ and, therefore, fi(f) 6 l, for every i 2 I, i P i0, that is the net {fi, i 2 I} fuzzy s-continuously converges to f.
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