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Arzela-Ascoli Theorems for Function Spaces with a Quasi-Uniform Range

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Arzela-Ascoli Theorems for Function Spaces with a Quasi-Uniform Range Journal of Mathematical Analysis ISSN: 2217-3412, URL: www.ilirias.com/jma Volume 8 Issue 5 (2017), Pages 97-104. ARZELA-ASCOLI THEOREMS FOR FUNCTION SPACES WITH A QUASI-UNIFORM RANGE LIAQAT ALI KHAN AND WAFA KHALAF ALQURASHI Abstract. Let X and Y be two topological spaces, F (X; Y ) the set of all functions from X into Y and C(X; Y ) the set of all continuous functions in F (X; Y ). In the case of Y = (Y; U), a uniform space, various uniform con- vergence topologies (such as UX ; Uk; Up) on F (X; Y ) and C(X; Y ) were sys- tematically studied by Kelley ([7], Chapter 7). Let S(X; Y ) (resp. P (X; Y ), L(X; Y )) denote the subspace of C(X; Y ) consisting of functions of all strongly continuous (resp. perfectly continuous, cl-supercontinuous) functions from X into Y . In the setting of (Y; U) a quasi-uniform space, we extend some recent results of Kohli and Singh [9] and Kohli and Aggarwal [8] on UX -closedness and completeness of L(X; Y ) in C(X; Y ), and also on Up-closedness and com- pactness of L(X; Y ) in C(X; Y ). In doing so, we first need to extend several supporting results of Kelley ([7], Chapter 7) on joint continuity, equicontinuity and even continuity from the setting of uniform to quasi-uniform spaces. 1. Introduction It is well known [7] that, if X is a topological space and Y = (Y; U) a uniform space, then (i) C(X; Y ) is UX -closed in F (X; Y ) ([7], Theorem 10, p. 228{229), (ii) but C(X; Y ) is not necessarily Up-closed in F (X; Y ) ([7], p. 229). In the case of Y a topological space, let tk (resp. tp) denote the compact-open (resp. point- open) topology on C(X; Y ) and F (X; Y ). In [16], Naimpally showed that if X is locally connected and Y is Hausdorff, then S(X; Y ) is tp-closed in F (X; Y ). Later more classes of variants of strong continuity were introduced by other authors, including the function spaces P (X; Y ) and L(X; Y ) having the relation: S(X; Y ) ⊆ P (X; Y ) ⊆ L(X; Y ) ⊆ C(X; Y ) ⊆ F (X; Y ). In particular, Kohli and Singh [9] extended Naimpally's result to a larger framework where X is a sum connected topological space and Y a Hausdorff space. In this case, L(X; Y ) = P (X; Y ) = S(X; Y ) and L(X; Y ) is tp-closed in F (X; Y ). Further, under some additional hypotheses including (Y; U) a Hausdorff uniform space, they obtained a criteria for Up-compactness of L(X; Y ) in C(X; Y ). The purpose of this paper to extend these results on closedness, completeness and compactness in C(X; Y ) in the setting of quasi-uniform topologies UX ; Uk and Up. 2000 Mathematics Subject Classification. 54C35, 54E15, 54C08. Key words and phrases. quasi-uniform convergence topology, Arzela-Ascoli Theorem, strongly continuous function, sum connected space, locally symmetric quasi-uniform space. c 2017 Ilirias Research Institute, Prishtin¨e,Kosov¨e. Submitted September 9, 2017. Published October 9, 2017. Communicated by Mati Abel. 97 98 L. A. KHAN AND W. K. ALQURASHI 2. Preliminaries We first recall necessary background for quasi-uniform spaces [3, 10, 14] and related concepts. Let Y be a non-empty set. A filter U on Y × Y is called a quasi-uniformity on Y if it satisfies the following conditions: (QU1) 4(Y ) = f(y; y): y 2 Y g ⊆ U for all U 2 U. 2 2 (QU2) If U 2 U, there is some V 2 U such that V ⊆ U. (Here V = V ◦ V = f(x; y) 2 Y × Y : 9 z 2 Y such that (x; z) 2 V and (z; y) 2 V g:) In this case, the pair (Y; U) is called a quasi-uniform space. If, in addition, U satisfies the symmetry condition: −1 (U3) U 2 U implies U := f(y; x):(x; y) 2 Ug 2 U, then U is called a uniformity on Y and the pair (Y; U) is called a uniform space. The pair (Y; U) is called a semi-uniform space [21] if U satisfies (QU1) and (U3). For any y 2 Y and U ⊆ Y × Y , let U[y] = fz 2 Y :(y; z) 2 Ug. A quasi-uniform space (Y; U) is called locally symmetric [3, 15] if, for each y 2 Y and each U 2 U, there is a symmetric V 2 U such that V 2[y] ⊆ U[y] (or equivalently [16], for any fixed integer n ≥ 1;there is a symmetric V 2 U such that V n[y] ⊆ U[y]). A semi- uniform space (Y; U) is called locally uniform [21] if, for each y 2 Y and each U 2 U, there is a V 2 U such that V 2[y] ⊆ U[y]. Clearly, every locally uniform space (Y; U) is a locally symmetric quasi-uniform space. If (Y; U) is a quasi-uniform space or a locally uniform space, then the collection T (U) = fG ⊆ Y : for each y 2 G, there is U 2 U such that U[y] ⊆ Gg is a topology, called the topology induced by U on Y ([3], p. 2{3; [21], p. 436). A net fyα : α 2 Dg in a quasi-uniform space (Y; U) is called a right K-Cauchy net provided that, for each U 2 U, there exists some α0 2 D such that (yα; yβ) 2 U for all α; β 2 D with α ≥ β ≥ α0. (Y; U) is called right K-complete if each right K-Cauchy net is T (U)-convergent in Y (cf. [11], Lemma 1, p. 289). Let X be a topological space and (Y; U) a quasi-uniform space, and let A = A(X) be a certain collection of subsets of X which covers X. For any A 2 A and U 2 U, let MA;U = f(f; g) 2 C(X; Y ) × C(X; Y ):(f(x); g(x)) 2 U for all x 2 Ag: Then the collection fMA;U : A 2 A and U 2 Ug forms a subbase for a quasi- uniformity, called the quasi-uniformity of quasi-uniform convergence on the sets in A induced by U. The resultant topology on C(X; Y ) is called the topology of quasi-uniform convergence on the sets in A and is denoted by UA [11]. (1) If A = fXg, UA is called the quasi-uniform topology of uniform con- vergence on C(X; Y ) and is denoted by UX . (2) If A = K(X)=fK ⊆ X : K is compactg, UA is called the quasi-uniform topology of compact convergence on F (X; Y ) and is denoted by Uk. (3) If A = F (X)=fA ⊆ X : A is finiteg, UA is called the quasi-uniform topology of pointwise convergence on C(X; Y ) and is denoted by Up. ARZELA-ASCOLI THEOREMS 99 Since each of the collection A in (1){(3) is closed under finite unions, the col- lection fMA;U : A 2 A and U 2 Ug actually forms a base for the topology UA (cf. [13], p. 7). Clearly, Up ⊆ Uk ⊆ UX on C(X; Y ). 3. UX -Closedness and Completeness in C(X,Y) In this section, we consider UX -closedness and right UX -K-completeness of L(X; Y ) in C(X; Y ). If X and Y are topological spaces, then a function f : X ! Y is said to be cl-supercontinuous [20] (=clopen-continuous [19]) if, for each open set H ⊆ Y containing f(x), there exists a clopen set G containing x in X such that f(G) ⊆ H. Let L(X; Y ) denote the set of all cl-supercontinuous functions from X into Y . It is easy to see that L(X; Y ) ⊆ C(X; Y ). It is shown in ([15], Theorem 3.5; [11], Lemma 2) that, if Y is a locally symmetric quasi-uniform space, C(X; Y ) is UX - closed in F (X; Y ). In this section we prove an analogous result for L(X; Y ). The following result extends ([8], Theorem 3.4) from the setting of uniform spaces to locally symmetric quasi-uniform spaces. We shall require the following result which extends ([7], Theorem 8, p. 226{227) from uniform to quasi-uniform spaces. Lemma 3.1. [1] Let X be a topological space and (Y; U) a quasi-uniform space. Let ffα : α 2 Dg be a net in F (X; Y ) such that: (i) ffα : α 2 Dg is a right K-Cauchy net in (F (X; Y ); UX ), Up (ii) fα −! f on X. UX Then fα −! f. Theorem 3.2. Let X be a topological space and (Y; U) a locally symmetric quasi-uniform space. Then (a) L(X; Y ) is UX -closed in both F (X; Y ) and C(X; Y ). (b) If Y is right K-complete, then L(X; Y ) is right UX -K-complete. Proof. (a) Let f 2 F (X; Y ) with f 2 UX -cl(L(X; Y )). To show that f 2 L(X; Y ), let x0 2 X and suppose that H is any open neighborhood of f(x0) in Y . We need to show that there exists a clopen set G containing x0 in X such that f(G) ⊆ H. Since H is open in Y , there exists U 2 U such that U[f(x0)] ⊆ H.
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