Novi Sad J. Math. Vol. 39, No. 1, 2009, 69-80 ON SOME CONVERGENCES FOR NETS OF FUNCTIONS WITH VALUES IN GENERALIZED UNIFORM SPACES T. B^³nzar1 Abstract. In this paper, we consider the almost uniform convergence and quasi-uniform and almost quasi-uniform ones for nets of functions with values in generalized uniform spaces. For such nets, the continuity and quasi-continuity of pointwise limit are studied. AMS Mathematics Subject Classi¯cation (2000): 54E15, 54E35, 26E25 Key words and phrases: generalized uniform spaces, continuous functions, quasi-continuous functions, almost uniform convergence, quasi-uniform convergence, almost quasi-uniform convergence 1. Introduction In [4] the quasi-uniform convergence in the sense of P.S. Alexandrov [1] was de¯ned for nets of functions with values in uniform spaces and some character- izations of the continuity of pointwise limit for such nets were given. The purpose of the present paper is to extend these results and the quasi- continuity property in the sense of Kempisty [8] to the nets of functions with values in generalized uniform spaces. 2. Preliminaries Let X be a nonempty set and ®; ¯ coverings of X. Following J. W. Tuckey [15] and K. Morita [10], ® is called a re¯nement of the covering ¯ (denoted ® Á ¯) i® for each[A 2 ® there exists B 2 ¯ such that A ½ B. By St(x; ®) is denoted the union fA 2 ® : x 2 Ag and it is called the star of x with respect to ®. For M ½ X the star of M with respect to ® is the set [ [ St(M; ®) = St1(M; ®) = fA 2 ® : A \ M 6= ;g = St(x; ®): x2M By recurrence we de¯ne Stn+1(M; ®) = St1(Stn(M; ®); ®) for n = 1; 2; :::. A family § of coverings of X is called a generalized uniform structure i® for any ®; ¯ 2 § there exists γ 2 § such that γ Á ®, γ Á ¯. Especially, § is called regular if: 1Department of Mathematics, "Politehnica" University of Timisoara, Piata Victoriei No. 2, 300006, Timisoara, ROMANIA, email: [email protected] 70 T. B^³nzar (a) For any x 2 X and for any ¯nite family fAigi=1;n with Ai 2 ®i 2 §, and \n \n x 2 Ai, there exists ¯ 2 § such that St(x; ¯) ½ Ai; i=1 i=1 (b) For any ® 2 § there exists ¯ 2 § with the property that for any B 2 ¯ there exist γB 2 § and AB 2 ® such that St(B; γB) ½ AB. If § is a generalized uniform structure for X, then (X; §) is called a gener- alized uniform space. For a generalized uniform space (X; §) we will use the topology ¿§ [14] determined by the subbase fA ½ X : A 2 ®; ® 2 §g. Let X and Y be topological spaces. Following S. Marcus [9] a function f : X ! Y is said to be quasi-continuous at x 2 X if for each neighbourhood U of x and each neighbourhood V of f(x) there exists a nonempty open set, G ½ U such that f(G) ½ V . The function f is said to be quasi-continuous on X if it is quasi-continuous at each x 2 X. Lemma 2.1. Fix a natural number, n ¸ 1. Then a function f de¯ned from a topological space (X; ¿) into a regular generalized uniform space (Y; §) is quasi- continuous if and only if for any (x0; ®) 2 X £ § and for each neighbourhood V n of x0, there exists a nonempty open set G ½ V such that f(G) ½ St (f(x0); ®). Proof. Suppose that f is quasi-continuous on X. Let (x0; ®) 2 X £ § and n V an arbitrary neighbourhood of x0. Then, since f(x0) 2 St (f(x0); ®) 2 ¿§, n there exists a nonempty open set G ½ V such that f(G) ½ St (f(x0); ®). Conversely, suppose that for any (x0; ®) 2 X £ §, and for every neigh- n bourhood V of x0 there exists G ½ V such that f(G) ½ St (f(x0); ®). Let x0 2 X and U 2 ¿§ with f(x0) 2 U. Then there exists ¯ 2 § such that n f(x0) 2 St (f(x0); ¯) ½ U (by Corollary 1.4 [5]). If V is an arbitrary neigh- bourhood of x0, then there exists a nonempty open set G ½ V such that n f(x) 2 St (f(x0); ¯) ½ U for every x 2 G, that is f is quasi-continuous at x0, whence f is quasi-continuous on X. 2 Remark 2.1. Analogously, it can be proved that if n 2 N, n ¸ 1, then a function f : X ! Y is continuous on X i® for any x0 2 X and ® 2 § there n exists a neighbourhood V of x0 such that f(V ) ½ St (f(x0); ®). Let (I; ¸), (J; ¸0) be two directed sets. Following J.L. Kelley [7], a function h : J ! I is said to be a "K-application" i® for each i 2 I there exists j 2 J such that for every j0 2 J, j0 ¸0 j it follows h(j0) ¸ i. 3. Almost uniform convergence Let (X; ¿) be a topological space, (Y; §) a generalized uniform space, and (I; ¸) a directed set. On some convergences for nets of functions... 71 De¯nition 3.1. [5] A net (fi)i2I of functions de¯ned on X into Y converges almost uniformly to a function f : X ! Y if for each x0 2 X and ® 2 § there exist i0 2 I and a neighbourhood V of x0 such that (1) fff(x); fi(x)g : x 2 V; i 2 I; i ¸ i0g Á ®: This de¯nition is equivalent to the following one: De¯nition 3.2. A net (fi)i2I , fi : X ! Y , converges almost uniformly to f : X ! Y if for each x0 2 X and ® 2 § there exist i0 2 I and a neighbourhood V of x0 such that (2) fi(x) 2 St(f(x); ®); for every x 2 V and i 2 I, i ¸ i0: Indeed, if relation (1) holds, then for i 2 I, i ¸ i0 and x 2 V it results that there exists A 2 ® such that ff(x); fi(x)g ½ A, that is fi(x) 2 St(f(x); ®). Conversely, if (2) holds, then for i 2 I, i ¸ i0 and x 2 V , there exists A 2 ® such that f(x) 2 A and fi(x) 2 A, that is ff(x); fi(x)g ½ A. Therefore (1) holds. It is known that the limit of an almost uniform convergent net of continuous functions is continuous too. Now, we give a similar result for the preservation of the quasi-continuity property, as follows: Theorem 3.1. Let (Y; §) be a regular generalized uniform space. If a net (fi)i2I of quasi-continuous functions de¯ned from X into Y converges almost uniformly to a function f : X ! Y , then f is quasi-continuous. Proof. Suppose that (fi)i2I converges almost uniformly to f : X ! Y and that fi is quasi-continuous on X for each i 2 I and let (x0; ®) 2 X £ §. Then there exist i0 2 I and V1 a neighbourhood of x0 such that (3) fi(x) 2 St(f(x); ®); for i 2 I, i ¸ i0 and x 2 V1: Let i 2 I, i ¸ i0, ¯xed and V an arbitrary neighbourhood of x0. Since fi is 0 quasi-continuous there exists a nonempty open set G ½ V = V1 \ V such that (4) fi(x) 2 St(fi(x0); ®); for x 2 G: Hence, using successively relations (3), (4), (3), we obtain: 2 3 f(x) 2 St(fi(x); ®) ½ St (fi(x0); ®) ½ St (f(x0); ®) for every x 2 G. Therefore, by Lemma 2.1, the quasi-continuity of f results. 2 In general, the converse of this theorem is not true. Thus, it is known that there exist examples of sequences of continuous real functions de¯ned on a compact set that converge pointwise to continuous functions but they do not converge uniformly [6]. Taking also into account that the almost uniform 72 T. B^³nzar convergence is equivalent to the uniform one on compact spaces [5] and that the continuity of functions implies the quasi-continuity, the statement is justi¯ed. In the next theorem we shall show how the almost uniform convergence char- acterizes the quasi-continuity of pointwise limit under some conditions which do not suppose by all means the quasi-continuity of net's terms. Theorem 3.2. If a net (fi)i2I of functions de¯ned on X into a regular general- ized uniform space (Y; §) converges almost uniformly to a function f : X ! Y , then f is quasi-continuous on X if and only if for any (x0; ®; i) 2 X £ § £ I there exists i1 2 I, i1 ¸ i such that, for each neighbourhood V of x0, there exists a nonempty open set G ½ V with the property (5) fi1 (x) 2 St(fi1 (x0); ®) ; for every x 2 G: Proof. Suppose that (fi)i2I converges almost uniformly on X to a quasi- continuous function f : X ! Y . Let (x0; ®; i) 2 X £ § £ I. Then, by Corollary 2 1.2 [5] there exist ¯0 2 § and A0 2 ® such that St (f(x0); ¯0) ½ A0. From the almost uniform convergence of the net (fi)i2I to f, it results that there exist i0 2 I and a neighbourhood V1 of x0 such that (6) f¹{(x) 2 St(f(x); ¯0); for every x 2 V1 and ¹{ 2 I, ¹{ ¸ i0: Let V be an arbitrary neighbourhood of x0. Then, since f is quasi-continuous, there exists a nonempty open set G ½ V \ V1 such that (7) f(x) 2 St(f(x0); ¯0) ; for every x 2 G: Now, let i1 2 I, i1 ¸ i, i1 ¸ i0.