GaGeoJaMo.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 6, No. 2, 2005 pp. 119-133 On some applications of fuzzy points M. Ganster, D. N. Georgiou, S. Jafari and S. P. Moshokoa Abstract. The notion of preopen sets (see [9] and [14]) play a very important role in General Topology and Fuzzy Topology. Preopen sets are also called nearly open and locally dense (see [4]). The purpose of this paper is to give some applications of fuzzy points in fuzzy topo- logical spaces. Moreover, in section 2 we offer some properties of fuzzy preclosed sets through the contribution of fuzzy points and we intro- duce new separation axioms in fuzzy topological spaces. Also using the notions of weak and strong fuzzy points, we investigate some proper- ties related to the preclosure of such points, and also their impact on separation axioms. In section 3, using the notion of fuzzy points, we introduce and study the notions of fuzzy pre-upper limit, fuzzy pre- lower limit and fuzzy pre-limit. Finally in section 4, we introduce the fuzzy pre-continuous convergence on the set of fuzzy pre-continuous functions and give a characterization of the fuzzy pre-continuous con- vergence through the assistance of fuzzy pre-upper limit. 2000 AMS Classification: 54A40 Keywords: Fuzzy Topology, Fuzzy points, Fuzzy convergence, Fuzzy sepa- ration axioms, Fuzzy preopen sets. 1. Introduction. Throughout this paper, the symbol I will denote the unit interval [0, 1]. In 1965, Zadeh (see [18]) introduced the fundamental notion of fuzzy set by which fuzzy mathematics emerged. Let X be a nonempty set. A fuzzy set in X is a function with domain X and values in I, i.e. an element of IX. A member A of IX is contained in a member B of IX, denoted by A ≤ B, if A(x) ≤ B(x) for every x ∈ X (see [18]). Let A, B ∈ IX. We define the following fuzzy sets (see [18]): (1) A ∧ B ∈ IX by (A ∧ B)(x) = min{A(x), B(x)} for every x ∈ X. (2) A ∨ B ∈ IX by (A ∨ B)(x) = max{A(x), B(x)} for every x ∈ X. (3) Ac ∈ IX by Ac(x) = 1 − A(x) for every x ∈ X. 120 M. Ganster, D. N. Georgiou, S. Jafari and S. P. Moshokoa (4) Let f : X → Y , A ∈ IX and B ∈ IY . Then f(A) is a fuzzy set in Y such that f(A)(y) = sup{A(x) : x ∈ f−1(y)}, if f−1(y) 6= ∅ and f(A)(y) = 0, if f−1(y) = ∅. Also, f−1(B) is a fuzzy set in X, defined by f−1(B)(x) = B(f(x)), x ∈ X. The first definition of a fuzzy topological space is due to Chang (see [3]). According to Chang, a fuzzy topological space is a pair (X, τ), where X is a set and τ is a fuzzy topology on it, i.e. a family of fuzzy sets (τ ⊆ IX) satisfying the following three axioms: (1) 0̄, 1̄ ∈ τ. By 0̄ and 1̄ we denote the characteristic functions X∅ and XX, respectively. (2) If A, B ∈ τ, then A ∧ B ∈ τ. (3) If {Aj : j ∈ J} ⊆ τ, then ∨{Aj : j ∈ J} ∈ τ. By using the notion of fuzzy set, Wong (see [15]) was able to introduce and investigate the notions of fuzzy points. In this paper we adopted Pu’s definition of a fuzzy point. A fuzzy set in a set X is called a fuzzy point if it takes the value 0 for all y ∈ X except one, say, x ∈ X. If its value at x is λ (0 < λ ≤ 1) we denote the fuzzy point by pλx, where the point x is called its support, denoted by supp(pλx), that is supp(p λ x) = x. The class of all fuzzy points in X is denoted by X . The fuzzy point pλx is said to be contained in a fuzzy set A or to belong to A, denoted by pλx ∈ A, if λ ≤ A(x). Evidently, every fuzzy set A can be expressed as the union of all the fuzzy points which belongs to A (see [10]). A fuzzy point pλx is said to be quasi-coincident with A denoted by p λ xqA if and only if λ > Ac(x) or λ + A(x) > 1 (see [10]). A fuzzy set A is said to be quasi-coincident with B, denoted AqB, if and only if there exists x ∈ X such that A(x) > Bc(x) or A(x) + B(x) > 1 (see [10]). If A does not quasi-coincident with B, then we write A q6 B. Let f be a function from X to Y . Then (see for example [1], [2], [3], [8], [11], [12], [13], [16], and [17]): (1) f−1(Bc) = (f−1(B))c, for any fuzzy set B in Y . (2) f(f−1(B)) ≤ B, for any fuzzy set B in Y . (3) A ≤ f−1(f(A)), for any fuzzy set A in X. (4) Let p be a fuzzy point of X, A be a fuzzy set in X and B be a fuzzy set in Y . Then, we have: (i) If f(p) q B, then p q f−1(B). (ii) If p q A, then f(p) q f(A). (5) Let A and B be fuzzy sets in X and Y , respectively and p be a fuzzy point in X. Then we have: (i) p ∈ f−1(B) if f(p) ∈ B. (ii) f(p) ∈ f(A) if p ∈ A. Let Λ be a directed set and X be an ordinary set. The function S : Λ → X is called a fuzzy net in X. For every λ ∈ Λ, S(λ) is often denoted by sλ and hence a net S is often denoted by {sλ, λ ∈ Λ} (see [10]). On some applications of fuzzy points 121 Let {An, n ∈ N} be a net of fuzzy sets in a fuzzy topological space X. Then by F − lim N (An), we denote the fuzzy upper limit of the net {An, n ∈ N} in IX, that is, the fuzzy set which is the union of all fuzzy points pλx in X such that for every n0 ∈ N and for every fuzzy open Q−neighborhood U of p λ x in X there exists an element n ∈ N for which n ≥ n0 and AnqU. In other cases we set F − lim N (An) = 0. For the notions of fuzzy upper limit and fuzzy lower limit see [6]. Recall that a fuzzy subset A of a fuzzy topological space X is called fuzzy preopen (see [5] and [14]) if A ≤ Int(Cl(A)), where Int and Cl denoted the interior and closure operators. A is called fuzzy preclosed if Cl(Int(A)) ≤ A. We denote the family of all fuzzy preopen (respectively, fuzzy preclosed) sets of X by FPO(X) (respectively, FPC(X)). Also the intersection of all fuzzy preclosed sets containing A is called fuzzy preclosure of A, denoted by pCl(A), that is pCl(A) = inf{K : A ≤ K, K ∈ FPC(X)}. Similar the fuzzy preinterior of A, denoted by pInt(A), is defined as follows: pInt(A) = sup{U : U ≤ A, U ∈ FPO(X)}. Let A be a fuzzy preopen (respectively, preclosed) set of a fuzzy space X. Then, by Theorem 3.7 of [14], pInt(A) = A (respectively, pCl(A) = A). Also, by Theorem 3.6 of [14], we have pCl(Ac) = 1̄ − pInt(A) = 1̄ − A = Ac (respec- tively, pInt(Ac) = 1̄ − pCl(A) = 1̄ − A = Ac). Thus, the fuzzy set Ac is fuzzy preclosed (respectively, preopen). 2. Fuzzy points, preclosed sets and separations axioms Definition 2.1. A fuzzy set A in a fuzzy space X is called a fuzzy pre- neighborhood of a fuzzy point pλx if there exists a V ∈ FPO(X) such that pλx ∈ V ≤ A. A fuzzy pre-neighborhood A is said to be preopen if A ∈ FPO(X). Definition 2.2. A fuzzy set A in a fuzzy space X is called a fuzzy Q−pre- neighborhood of pλx if there exists B ∈ FPO(X) such that p λ xqB and B ≤ A. Remark 2.3. A fuzzy Q−pre-neighborhood of a fuzzy point generally does not contain the point itself. In what follows by NQ−p−n(p λ x) we denote the family of all fuzzy preopen Q−pre-neighborhoods of the fuzzy point pλx in X. The set NQ−p−n(p λ x) with the relation ≤ ∗ (that is, U1 ≤ ∗ U2 if and only if U2 ≤ U1) form a directed set. Proposition 2.4. Let A be a fuzzy set of a fuzzy space X. Then, a fuzzy point pλx ∈ pCl(A) if and only if for every U ∈ FPO(X) for which p λ xqU we have UqA. Proof. The fuzzy point pλx ∈ pCl(A) if and only if p λ x ∈ F , for every fuzzy preclosed set F of X for which A ≤ F . Equivalently pλx ∈ pCl(A) if and only if λ ≤ 1 − U(x), for every fuzzy preopen set U for which A ≤ 1̄ − U. Thus 122 M. Ganster, D. N. Georgiou, S. Jafari and S. P. Moshokoa pλx ∈ pCl(A) if and only if U(x) ≤ 1 − λ, for every fuzzy preopen set U for which U ≤ 1̄ − A. So, pλx ∈ pCl(A) if and only if for every fuzzy preopen set U of X such that U(x) > 1 − λ we have U 6≤ 1̄ − A. Therefore by Proposition 2.1 of [10], pλx ∈ pCl(A) if and only if for every fuzzy preopen set U of X such that U(x) + λ > 1 we have UqA. Thus, pλx ∈ pCl(A) if and only if for every fuzzy preopen set U of X such that pλxqU we have UqA. � Definition 2.5. Let A be a fuzzy set of a fuzzy space X. A fuzzy point pλx is called a pre-boundary point of a fuzzy set A if and only if pλx ∈ pCl(A) ∧ (1̄ − pCl(A)). By pBd(A) we denote the fuzzy set pCl(A) ∧ (1̄ − pCl(A)). Proposition 2.6. Let A be a fuzzy set of a fuzzy space X. Then A ∨ pBd(A) ≤ pCl(A). Proof. Let pλx ∈ A ∨ pBd(A). Then p λ x ∈ A or p λ x ∈ pBd(A). Clearly, if pλx ∈ pBd(A), then p λ x ∈ pCl(A). Let us suppose that p λ x ∈ A. We have pCl(A) = ∧{F : F ∈ IX, F is preclosed and A ≤ F}. So, if pλx ∈ A, then p λ x ∈ F , for every fuzzy preclosed set F of X for which A ≤ F and therefore pλx ∈ pCl(A). � Example 2.7. Let (X, τ) be a fuzzy space such that X = {x, y} and τ = {0̄, 1̄, p 1 2 x }. The family of all fuzzy preclosed sets of X contains the following fuzzy sets A of X: i) A ∈ IX such that A(x) ∈ [0, 1 2 ) and A(y) ∈ [0, 1]. Indeed, Cl(Int(A)) = Cl(0̄) = 0̄ ≤ A. ii) A ∈ IX such that A(x) ∈ [1 2 , 1] and A(y) = 1. Indeed, Cl(Int(A)) = Cl(p 1 2 x ) ≤ (p 1 2 x ) c ≤ A. Also, the family of all fuzzy preopen sets of X are the following fuzzy sets U of X: i) U ∈ IX such that U(x) ∈ [0, 1 2 ] and U(y) = 0. Indeed, Int(Cl(U)) = Int((p 1 2 x ) c) = p 1 2 x ≥ U. ii) U ∈ IX such that U(x) ∈ (1 2 , 1] and U(y) ∈ [0, 1]. Indeed, Int(Cl(U)) = Int(1̄) = 1̄ ≥ U. We consider the fuzzy set B ∈ IX such that B = p 2 3 x . By the above we have: pCl(B) = (p 1 3 x ) c, where (p 1 3 x ) c(z) = 2 3 , if z = x and (p 1 3 x ) c(z) = 1, if z = y. On some applications of fuzzy points 123 Also, we have 1̄ − pCl(B) = p 1 3 x and pBd(B) = pCl(B) ∧ (1̄ − pCl(B)) = p 1 3 x . Thus B ∨ pBd(B) = B 6= pCl(B). Definition 2.8. A fuzzy space X is called pre-T0 if for every two fuzzy points pλx and p µ y such that p λ x 6= p µ y , either p λ x 6∈ pCl(p µ y) or p µ y 6∈ pCl(p λ x). Definition 2.9. A fuzzy space X is called pre-T1 if every fuzzy point is fuzzy preclosed. Remark 2.10. Clearly, every pre-T1 fuzzy space is pre-T0. Proposition 2.11. A fuzzy space X is pre-T1 if and only if for each x ∈ X and each λ ∈ [0, 1] there exists a fuzzy preopen set A such that A(x) = 1 − λ and A(y) = 1 for y 6= x. Proof. ⇒) Let λ = 0. We set A = 1̄. Then A is fuzzy preopen set such that A(x) = 1 − 0 and A(y) = 1 for y 6= x. Now, let λ ∈ (0, 1] and x ∈ X. We set A = (pλx) c. The set A is fuzzy preopen such that A(x) = 1 − λ and A(y) = 1 for y 6= x. ⇐) Let pλx be an arbitrary fuzzy point of X. We prove that the fuzzy point pλx is fuzzy preclosed. By assumption there exists a fuzzy preopen set A such that A(x) = 1 − λ and A(y) = 1 for y 6= x. Clearly, Ac = pλx. Thus the fuzzy point pλx is fuzzy preclosed and therefore the fuzzy space X is pre-T1. � Definition 2.12. A fuzzy space X is called a pre-Hausdorff space if for any fuzzy points pλx and p µ y for which supp(p λ x) = x 6= supp(p µ y) = y, there exist two fuzzy preopen Q-pre-neighbourhoods U and V of pλx and p µ y, respectively, such that U ∧ V = 0̄. Example 2.13. Let (X, τ) be a fuzzy space such that X = {x, y} and τ = {0̄, 1̄, p 1 2 x }. The fuzzy point p 1 2 x is not fuzzy preclosed. Indeed, we have: Cl(Int(p 1 2 x )) = Cl(p 1 2 x ) = (p 1 2 x ) c 6≤ p 1 2 x . Thus the fuzzy space X is not pre-T1. Also, it is clear that the fuzzy space X is pre-T0. Example 2.14. Let (X, τ) be a fuzzy space such that X = {x, y} and τ = {0̄, 1̄}. We observe that every fuzzy point pλx is fuzzy preclosed. Indeed, we have Cl(Int(pλx)) = 0̄ ≤ p λ x. Thus the fuzzy space X is pre-T1 and therefore is pre-T0. Also, it is clear that the fuzzy space X is pre-Hausdorff. 124 M. Ganster, D. N. Georgiou, S. Jafari and S. P. Moshokoa It is not difficult to see that the fuzzy space X is not T0, T1 and Hausdorff. For the definitions of T0, T1 and Hausdorff fuzzy spaces see [10]. Definition 2.15. A fuzzy space X is called a pre-regular space if for any fuzzy point pλx and a fuzzy preclosed set F not containing p λ x, there exist U, V ∈ FPO(X) such that pλx ∈ U, F ≤ V and U ∧ V = 0̄. Example 2.16. Let (X, τ) be a fuzzy space such that X = {x, y} and τ = {0̄, 1̄}. The fuzzy space X is pre-Hausdorff but it is not pre-regular. We prove only that the fuzzy space X is not pre-regular. We consider the fuzzy point p 1 3 x and the fuzzy set A of X such that A(x) = 1 4 and A(y) = 1. For the fuzzy set A we have Cl(Int(A)) = 0̄ ≤ A. Thus the fuzzy set A is fuzzy preclosed. Also, we have p 1 3 x 6∈ A. If U and V are two arbitrary fuzzy preopen sets such that p 1 3 x ∈ U and A ≤ V , then (U ∧ V )(x) ≥ 1 4 and therefore U ∧ V 6= 0̄. Thus the fuzzy space X is not pre-regular. Definition 2.17. A fuzzy space X is called a quasi pre-T1 if for any fuzzy points pλx and p µ y for which supp(p λ x) = x 6= supp(p µ y) = y, there exists a fuzzy preopen set U such that pλx ∈ U and p µ y 6∈ U and another V such that p λ x 6∈ V and pµy ∈ V . Example 2.18. Let (X, τ) be a fuzzy space such that X = {x, y} and τ = {0̄, 1̄, p 1 2 x }. The fuzzy space X is quasi pre-T1 but it is not pre-T1. Definition 2.19. (see [7]) A fuzzy point pλx is called weak (respectively, strong) if λ ≤ 1 2 (respectively, λ > 1 2 ). Definition 2.20. A fuzzy set A of a fuzzy space X is called pre-generalized closed (briefly fpg-closed) if pCl(A) ≤ U whenever A ≤ U and U fuzzy preopen set of X. Proposition 2.21. Let X be a fuzzy space X. Suppose that pλx and p µ y are weak and strong fuzzy points, respectively. If pλx is pre-generalized closed, then pµy ∈ pCl(p λ x) ⇒ p λ x ∈ pCl(p µ y). Proof. Suppose that pµy ∈ pCl(p λ x) and p λ x 6∈ pCl(p µ y). Then pCl(p µ y)(x) < λ. Also λ ≤ 1 2 . Thus pCl(pµy)(x) ≤ 1 − λ and therefore λ ≤ 1 − pCl(p µ y)(x). So pλx ∈ (pCl(p µ y )) c. But pλx is pre-generalized closed and (pCl(p µ y )) c is fuzzy preopen. Thus pCl(pλx) ≤ (pCl(p µ y)) c . By assumption we have pµy ∈ pCl(p λ x). Thus pµy ∈ (pCl(p µ y)) c. On some applications of fuzzy points 125 We prove that this is a contradiction. Indeed, we have µ ≤ 1 − pCl(pµy )(y) or pCl(pµy)(y) ≤ 1 − µ. Also pµy ∈ pCl(p µ y). Thus µ ≤ 1 − µ. But pµy is a strong fuzzy point, that is µ > 1 2 . So the above relation µ ≤ 1−µ is a contradiction. Thus pλx ∈ pCl(p µ y). � Proposition 2.22. If X is a quasi pre-T1 fuzzy space and p λ x a weak fuzzy point in X, then (pλx) c is a fuzzy pre-neighborhood of each fuzzy point pµy with y 6= x. Proof. Let y 6= x and pµy be a fuzzy point of X. Since the space X is a quasi pre-T1 there exists a fuzzy preopen U of X such that p µ y ∈ U and p λ x 6∈ U. This implies that λ > U(x). Also, λ ≤ 1 2 . Thus U(x) ≤ 1 − λ. Therefore U(y) ≤ 1 = (pλx) c(y), for every y ∈ X \ {x}. So U ≤ (pλx) c. Therefore the fuzzy point pλx is a pre-neighborhood of p µ y . � Proposition 2.23. If X is a pre-regular fuzzy space, then for any strong fuzzy point pλx and any fuzzy preopen set U containing p λ x, there exists a fuzzy preopen set W containing pλx such that pCl(W) ≤ U. Proof. Suppose that pλx is any strong fuzzy point contained in U ∈ FPO(X). Then 1 2 < λ ≤ U(x). Thus the complement of U, that is the fuzzy set Uc, is a fuzzy preclosed set to which does not belong the fuzzy point pλx. Thus, there exist W, V ∈ FPO(X) such that pλx ∈ W and U c ≤ V with W ∧ V = 0̄. Hence, we have W ≤ V c and by Theorem 3.8 of [14] pCl(W) ≤ pCl(V c) = V c. Now Uc ≤ V implies V c ≤ U. This means that pCl(W) ≤ U which completes the proof. � Proposition 2.24. If X is a fuzzy pre-regular space, then the strong fuzzy points in X are fpg-closed. Proof. Let pλx be any strong fuzzy point in X and U be a fuzzy open set such that pλx ∈ U. By Proposition 2.23 there exists a W ∈ FPO(X) such that pλx ∈ W and pCl(W) ≤ U. By Theorem 3.8 of [14], we have pCl(pλx) ≤ pCl(W) ≤ U. Thus the fuzzy point pλx is fpg-closed. � Definition 2.25. A fuzzy space X is called a weakly pre-regular space if for any weak fuzzy point pλx and a fuzzy preclosed set F not containing p λ x, there exist U, V ∈ FPO(X) such that pλx ∈ U, F ≤ V and U ∧ V = 0̄. Observe that every pre-regular fuzzy space is weakly pre-regular. 126 M. Ganster, D. N. Georgiou, S. Jafari and S. P. Moshokoa Definition 2.26. Let X be a fuzzy space. A fuzzy set U in X is said to be fuzzy pre-nearly crisp if pCl(U) ∧ (pCl(U))c = 0̄. Proposition 2.27. Let X be a fuzzy space. If for any weak fuzzy point pλx and any U ∈ FPO(X) containing pλx, there exists a fuzzy preopen and pre-nearly crisp fuzzy set W containing pλx such that pCl(W) ≤ U, then X is fuzzy weakly pre-regular. Proof. Assume that F is a fuzzy preclosed set not containing the weak fuzzy point pλx. Then F c is a fuzzy preopen set containing pλx. By hypothesis, there exists a fuzzy preopen and pre-nearly crisp fuzzy set W such that pλx ∈ W and pCl(W) ≤ F c. We set N = pInt(pCl(W)) and M = 1 − pCl(W). Then N is fuzzy preopen, pλx ∈ N and F ≤ M. We are going to prove that M ∧ N = 0̄. Now assume that there exists y ∈ X such that (N ∧ M)(y) = µ 6= 0̄. Then pµy ∈ N ∧M. Hence, p µ y ∈ pCl(W) and p µ y ∈ (̄pCl(W)) c. This is a contradiction since W is pre-nearly crisp. Thus the fuzzy space X is weakly pre-regular. � Definition 2.28. Let X be a fuzzy space. A fuzzy point pλx in X is said to be well-preclosed if there exists pµy ∈ pCl(p λ x) such that supp(p λ x) 6= supp(p µ y). Proposition 2.29. If X is a fuzzy space and pλx is a fpg-closed, well-preclosed fuzzy point, then X is not quasi pre-T1 space. Proof. Let X be a fuzzy quasi pre-T1 space. By the fact p λ x is well-preclosed, there exists a fuzzy point pµy with supp(p λ x) 6= supp(p µ y) such that p µ y ∈ pCl(p λ x). Then there exists U ∈ FPO(X) such that pλx ∈ U and p µ y 6∈ U. Therefore pCl(pλx) ≤ U and p µ y ∈ U. But this is a contradiction and hence X can not be quasi pre-T1 space. � Definition 2.30. Let X be a fuzzy space. A fuzzy point pλx is said to be just- preclosed if the fuzzy set pCl(pλx) is again fuzzy point. Clearly, in a fuzzy pre-T1 space every fuzzy point is just-preclosed. Proposition 2.31. Let X be a fuzzy space. If pλx and p µ x are two fuzzy points such that λ < µ and pµx is fuzzy preopen, then p λ x is just-preclosed if it is fpg- closed. Proof. We prove that the fuzzy set pCl(pλx) is again a fuzzy point. We have pλx ∈ p µ x and the fuzzy set p µ x is fuzzy preopen. Since p λ x is fpg-closed we have pCl(pλx) ≤ p µ x. Thus pCl(p λ x)(x) ≤ µ and pCl(p λ x)(z) ≤ 0, for every z ∈ X \{x}. So the fuzzy set pCl(pλx) is a fuzzy point. � 3. Fuzzy pre-convergence and fuzzy points Definition 3.1. Let {An, n ∈ N} be a net of fuzzy sets in a fuzzy space X. Then by F − pre − lim N (An), we denote the fuzzy pre-upper limit of the net {An, n ∈ N} in I X, that is, the fuzzy set which is the union of all fuzzy On some applications of fuzzy points 127 points pλx in X such that for every n0 ∈ N and for every fuzzy preopen Q−pre- neighborhood U of pλx in X there exists an element n ∈ N for which n ≥ n0 and AnqU. In other cases we set F − pre − lim N (An) = 0. Example 3.2. Let (X, τ) be a fuzzy space such that X = {x, y} and τ = {0̄, 1̄, p 1 2 x }. Also let {An, n ∈ N} be a net of fuzzy sets of X such that An(X) = {0.5} for every n ∈ N. The fuzzy point p 1 2 x ∈ F − lim N (An). Indeed, for every n0 ∈ N and for the only fuzzy open Q-neighborhood U = 1̄ of p 1 2 x there exists an element n ∈ N for which n ≥ n0 and AnqU. The fuzzy point p 1 2 x 6∈ F − pre − lim N (An). Indeed, for every n0 ∈ N and for the fuzzy preopen Q-pre-neighborhood U = p 2 3 x of p 1 2 x does not exist any element n ∈ N such that n ≥ n0 and AnqU. By the above we have F − lim N (An) 6= F − pre − lim N (An). Definition 3.3. Let {An, n ∈ N} be a net of fuzzy sets in a fuzzy space X. Then by F − pre − lim N (An), we denote the fuzzy pre-lower limit of the net {An, n ∈ N} in I X, that is, the fuzzy set which is the union of all fuzzy points pλx in X such that for every fuzzy preopen Q−pre-neighborhood U of p λ x in X there exists an element n0 ∈ N such that AnqU, for every n ∈ N, n ≥ n0. In other cases we set F − pre − lim N (An) = 0. Definition 3.4. A net {An, n ∈ N} of fuzzy sets in a fuzzy topological space X is said to be fuzzy pre-convergent to the fuzzy set A if F − pre − lim N (An) = F − pre − lim N (An) = A. We then write F − pre − lim N (An) = A. Proposition 3.5. Let {An, n ∈ N} and {Bn, n ∈ N} be two nets of fuzzy sets in X. Then the following statements are true: (1) The fuzzy pre-upper limit is preclosed. (2) F − pre − lim N (An) =F − lim N (pCl(An)). (3) If An = A for every n ∈ N, then F − pre − lim N (An) = pCl(A) (4) The fuzzy upper limit is not affected by changing a finite number of the An. (5) F − pre − lim N (An) ≤ pCl(∨{An : n ∈ N}). (6) If An ≤ Bn for every n ∈ N, then F −pre−lim N (An) ≤F −pre−lim N (Bn). (7) F − pre − lim N (An ∨ Bn) =F − pre − lim N (An)∨F − pre − lim N (Bn). (8) F − pre − lim N (An ∧ Bn) ≤F − pre − lim N (An)∧F − pre − lim N (Bn). Proof. We prove only the statements (1)-(5). (1) It is sufficient to prove that 128 M. Ganster, D. N. Georgiou, S. Jafari and S. P. Moshokoa pCl(F − pre − lim N (An)) ≤ F − pre − lim N (An). Let prx ∈ pCl(F − pre − lim N (An)) and let U be an arbitrary fuzzy preopen Q−pre-neighborhood of pry. Then, we have: UqF − pre − lim N (An). Hence, there exists an element x′ ∈ X such that U(x′) + F − pre − lim N (An)(x ′) > 1. Let F − pre − lim N (An)(y ′) = k. Then, for the fuzzy point pkx′ in X we have p k x′ q U and p k x′ ∈ F − pre − lim N (An). Thus, for every element n0 ∈ N there exists n ≥ n0, n ∈ N such that AnqU. This means that prx ∈ F − pre − lim N (An). (2) Clearly, it is sufficient to prove that for every fuzzy preopen set U the condition UqAn is equivalent to UqpCl(An). Let UqAn. Then there exists an element x ∈ X such that U(y)+An(x) > 1. Since An ≤ pCl(An) we have U(x)+pCl(An)(x) > 1 and therefore UqpCl(An). Conversely, let UqpCl(An). Then there exists an element x ∈ X such that U(x) + pCl(An)(x) > 1. Let pCl(An)(x) = r. Then p r x ∈ pCl(An) and the fuzzy preopen set U is a fuzzy preopen Q−pre-neighborhood of prx. Thus UqAn. (3) It follows by Proposition 2.4 and the definition of the fuzzy pre-upper limit. (4) It follows by definition of the fuzzy pre-upper limit. (5) Let prx ∈F−pre−lim N (An) and U be a fuzzy preopen Q−pre-neighborhood of prx in X. Then for every n0 ∈ N there exists n ∈ N, n ≥ n0 such that AnqU and therefore ∨{An : n ∈ N}qU. Thus, p r x ∈ pCl(∨{An : n ∈ N}). � Proposition 3.6. Let {An, n ∈ N} and {Bn, n ∈ N} be two nets of fuzzy sets in Y . Then the following statements are true: (1) The fuzzy pre-lower limit is preclosed. (2) F − pre − lim N (An) =F − pre − lim N (pCl(An)). (3) If An = A for every n ∈ N, then F − pre − lim N (An) = pCl(A) (4) The fuzzy upper limit is not affected by changing a finite number of the An. (5) ∧{An : n ∈ N} ≤F − pre − lim N (An). (6) F − pre − lim N (An) ≤ pCl(∨{An : n ∈ N}). (7) If An ≤ Bn for every n ∈ N, then F −pre−lim N (An) ≤F −pre−lim N (Bn). (8) F − pre − lim N (An ∨ Bn) ≥F − pre − lim N (An)∨F − pre − lim N (Bn). (9) F − pre − lim N (An ∧ Bn) ≤F − pre − lim N (An)∧F − pre − lim N (Bn). Proof. The proof is similar to the proof of Proposition 3.5. � On some applications of fuzzy points 129 Proposition 3.7. For the fuzzy upper and lower limit we have the relation F − pre − lim N (An) ≤ F − pre − lim N (An). Proof. It is a consequence of definitions of fuzzy pre-upper and fuzzy pre-lower limits. � Proposition 3.8. Let {An, n ∈ N} and {Bn, n ∈ N} be two nets of fuzzy sets in a fuzzy space Y . Then the following propositions are true (in the following properties the nets {An, n ∈ N} and {Bn, n ∈ N} are supposed to be fuzzy pre-convergent): (1) pCl(F − pre− lim N (An)) = F − pre− lim N (An) = F − pre− lim N (pCl(An)). (2) If An = A for every n ∈ N, then F − pre − lim N (An) = pCl(A) (3) If An ≤ Bn for every n ∈ N, then F −pre−lim N (An) ≤F −pre−lim N (Bn). (4) F − pre − lim N (An ∨ Bn) =F − pre − lim N (An)∨F − pre − lim N (Bn). Proof. The proof of this proposition follows by Propositions 3.5 and 3.6. � 4. Fuzzy pre-continuous functions, fuzzy pre-continuous convergence and fuzzy points Definition 4.1. A function f from a fuzzy space Y into a fuzzy space Z is called fuzzy pre-continuous if for every fuzzy point pλx in Y and every fuzzy preopen Q−pre-neighborhood V of f(pλx), there exists a fuzzy preopen Q−pre- neighborhood U of pλx such that f(U) ≤ V . Let Y and Z be two fuzzy spaces. Then by FPC(Y, Z) we denote the set of all fuzzy pre-continuous maps of Y into Z. Example 4.2. Let (Y, τ1) and (Y, τ2) be two fuzzy spaces such that Y = {x, y}, τ1 = {0̄, 1̄} and τ2 = {0̄, 1̄, p 1 2 x }. We consider the map i : (Y, τ1) → (Y, τ2) for which i(z) = z for every z ∈ Y . We prove that the map i is not fuzzy continuous at the fuzzy point p0.8x but it is fuzzy pre-continuous at the fuzzy point p0.8x . Indeed, for the fuzzy open Q-neighborhood V = p 1 2 x of i(p 0.8 x ) = p 0.8 x does not exist a fuzzy open Q-neighborhood U of p0.8x such that i(U) ≤ V . The only fuzzy open Q-neighborhood U of p0.8x in (Y, τ1) is the fuzzy set 1̄ and i(1̄) 6≤ V . Now, we prove that the map i is fuzzy pre-continuous at the fuzzy point p0.8x . Let V be an arbitrary fuzzy preopen Q−pre-neighborhood V of i(p0.8x ) = p 0.8 x . The family of all fuzzy preopen sets of (Y, τ2) are the following fuzzy sets V of Y : i) V ∈ IY such that V (x) ∈ [0, 1 2 ] and V (y) = 0. ii) V ∈ IY such that V (x) ∈ (1 2 , 1] and V (y) ∈ [0, 1]. The above fuzzy sets V (cases i) and ii)) are also fuzzy preopen sets of (Y, τ1). So for every fuzzy preopen Q−pre-neighborhood V of i(p0.8x ) in (Y, τ2) there exists the fuzzy preopen Q−pre-neighborhood U = V of p0.8x in (Y, τ1) such that i(U) ≤ V . 130 M. Ganster, D. N. Georgiou, S. Jafari and S. P. Moshokoa Definition 4.3. A fuzzy net S = {sλ, λ ∈ Λ} in a fuzzy space (X, τ) is said to be p-convergent to a fuzzy point e in X relative to τ and write p lim sλ = e if for every fuzzy preopen Q-pre-neighborhood U of e and for every λ ∈ Λ there exists m ∈ Λ such that Uqsm and m ≥ λ. Proposition 4.4. A function f from a fuzzy space X into a fuzzy space Y is fuzzy pre-continuous if and only if for every fuzzy net S = {sλ, λ ∈ Λ}, S p- converges to p, then f ◦ S = {f(sλ), λ ∈ Λ} is a fuzzy net in Y and p-converges to f(p). Proof. It is obvious. � Proposition 4.5. Let f : Y → Z be a fuzzy pre-continuous map, p be a fuzzy point in Y and U, V be fuzzy preopen Q−neighborhoods of p and f(p), respectively such that f(U) 6≤ V . Then there exists a fuzzy point p1 in Y such that p1qU and f(p1) q6 V . Proof. Since f(U) 6≤ V . We have U 6≤ f−1(V ). Thus there exists x ∈ Y such that U(x) > f−1(V )(x) or U(x) − f−1(V )(x) > 0 and therefore U(x) + 1 − f−1(V )(x) > 1 or U(x) + (f−1(V ))c(x) > 1. Let (f−1(V ))c(x) = r. Clearly, for the fuzzy point prx we have p r xqU and p r x ∈ (f −1(V ))c. Hence for the fuzzy point p1 ≡ p r x we have p1qU and f(p1) q6 V . � Definition 4.6. A net {fµ, µ ∈ M} in FPC(Y, Z) fuzzy pre-continuously converges to f ∈ FPC(Y, Z) if for every fuzzy net {pλ, λ ∈ Λ} in Y which p-converges to a fuzzy point p in Y we have that the fuzzy net {fµ(pλ), (λ, µ) ∈ Λ × M} p-converges to the fuzzy point f(p) in Z. Proposition 4.7. A net {fµ, µ ∈ M} in FPC(Y, Z) fuzzy pre-continuously converges to f ∈ FC(Y, Z) if and only if for every fuzzy point p in Y and for every fuzzy preopen Q−pre-neighborhood V of f(p) in Z there exist an element µ0 ∈ M and a fuzzy preopen Q−pre-neighborhood U of p in Y such that fµ(U) ≤ V, for every µ ≥ µ0, µ ∈ M. Proof. Let p be a fuzzy point in Y and V be a fuzzy preopen Q−pre-neighborhood of f(p) in Z such that for every µ ∈ M and for every fuzzy preopen Q−pre- neighborhood U of p in Y there exists µ′ ≥ µ such that fµ′(U) 6≤ V. Then for every fuzzy preopen Q−neighborhood U of p in Y we can choose a fuzzy point pU in Y (see Proposition 4.5) such that pU q U and fµ′(pU ) 6q V. Clearly, the fuzzy net {pU, U ∈ NQ−p−n(p)} p-converges to p, but the fuzzy net {fµ(pU ), (U, µ) ∈ NQ−p−n(p) × M} does not p-converge to f(p) in Z. On some applications of fuzzy points 131 Conversely, let {pλ, λ ∈ Λ} be a fuzzy net in FPC(Y, Z) which p-converges to the fuzzy point p in Y and let V be an arbitrary fuzzy preopen Q−pre- neighborhood of f(p) in Z. By assumption there exist a fuzzy preopen Q−pre- neighborhood U of p in Y and an element µ0 ∈ M such that fµ(U) ≤ V , for every µ ≥ µ0, µ ∈ M. Since the fuzzy net {pλ, λ ∈ Λ} p-converges to p in Y . There exists λ0 ∈ Λ such that pλqU, for every λ ∈ Λ, λ ≥ λ0. Let (λ0, µ0) ∈ Λ × M. Then for every (λ, µ) ∈ Λ × M, (λ, µ) ≥ (λ0, µ0) we have fµ(pλ) q fµ(U) ≤ V , that is fµ(pλ) q V . Thus the net {fµ(pλ), (λ, µ) ∈ Λ×M} p-converges to f(p) and the net {fµ, µ ∈ M} fuzzy pre-continuously converges to f. � Proposition 4.8. A net {fλ, λ ∈ Λ} in FPC(Y, Z) fuzzy pre-continuously converges to f ∈ FPC(Y, Z) if and only if F − pre − lim Λ (f−1 λ (K)) ≤ f−1(K), (1) for every fuzzy preclosed subset K of Z. Proof. Let {fλ, λ ∈ Λ} be a net in FPC(Y, Z), which fuzzy pre-continuously converges to f and let K be an arbitrary fuzzy preclosed subset of Z. Let p ∈F− pre − lim Λ (f−1 λ (K)) and W be an arbitrary fuzzy preopen Q−pre-neighborhood of f(p) in Z. Since the net {fλ, λ ∈ Λ} fuzzy pre-continuously converges to f, there exist a fuzzy preopen Q−pre-neighborhood V of p in Y and an element λ0 ∈ Λ such that fλ(V ) ≤ W , for every λ ∈ Λ, λ ≥ λ0. On the other hand, there exists an element λ ≥ λ0 such that V qf −1 λ (K). Hence, fλ(V )qK and therefore WqK. This means that f(p) ∈ pCl(K) = K. Thus, p ∈ f−1(K). Conversely, let {fλ, λ ∈ Λ} be a net in FPC(Y, Z) and f ∈ FPC(Y, Z) such that the relation (1) holds for every fuzzy preclosed subset K of Z. We prove that the net {fλ, λ ∈ Λ} fuzzy continuously converges to f. Let p be a fuzzy point of Y and W be a fuzzy preopen Q−pre-neighborhood of f(p) in Z. Since p 6∈ f−1(K), where K = W c we have p 6∈ F − pre − lim Λ (f−1 λ (K)). This means that there exist an element λ0 ∈ Λ and a fuzzy preopen Q−pre- neighborhood V of p in Y such that f−1 λ (K) q6 V , for every λ ∈ Λ, λ ≥ λ0. Then we have V ≤ (f−1 λ (K))c = f−1 λ (Kc) = f−1 λ (W). Therefore, fλ(V ) ≤ W , for every λ ∈ Λ, λ ≥ λ0, that is the net {fλ, λ ∈ Λ} fuzzy pre-continuously converges to f. � Proposition 4.9. The following statements are true: (1) If {fλ, λ ∈ Λ} is a net in FPC(Y, Z) such that fλ = f, for every λ ∈ Λ, then the {fλ, λ ∈ Λ} fuzzy pre-continuously converges to f ∈ FPC(Y, Z). 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Ganster (ganster@weyl.math.tu-graz.ac.at) Department of Mathematics, Graz University of Technology, Steyrergasse 30 A-8010 Graz, Austria. D. N. Georgiou (georgiou@math.upatras.gr) Department of Mathematics, University of Patras, 265 00 Patras, Greece. S. Jafari (sjafari@ruc.dk) Department of Mathematics and Physics, Roskilde University, Postbox 260, 4000 Roskilde, Denmark. S. P. Moshokoa (moshosp@unisa.ac.za) Department of Mathematics, Applied Mathematics and Astronomy, P. O. Box 392, Pretoria, 0003, South Africa.