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 convergence 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 compactness 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, Prishtinë,Kosovë. 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 ) = {(y, y): y ∈ Y } ⊆ U for all U ∈ U. 2 2 (QU2) If U ∈ U, there is some V ∈ U such that V ⊆ U. (Here V = V ◦ V = {(x, y) ∈ Y × Y : ∃ z ∈ Y such that (x, z) ∈ V and (z, y) ∈ V }.) In this case, the pair (Y, U) is called a quasi-uniform space. If, in addition, U satisfies the symmetry condition: −1 (U3) U ∈ U implies U := {(y, x):(x, y) ∈ U} ∈ 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 ∈ Y and U ⊆ Y × Y , let U[y] = {z ∈ Y :(y, z) ∈ U}. A quasi-uniform space (Y, U) is called locally symmetric [3, 15] if, for each y ∈ Y and each U ∈ U, there is a symmetric V ∈ U such that V 2[y] ⊆ U[y] (or equivalently [16], for any fixed integer n ≥ 1,there is a symmetric V ∈ U such that V n[y] ⊆ U[y]). A semi- uniform space (Y, U) is called locally uniform [21] if, for each y ∈ Y and each U ∈ U, there is a V ∈ 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) = {G ⊆ Y : for each y ∈ G, there is U ∈ U such that U[y] ⊆ G} is a topology, called the topology induced by U on Y ([3], p. 2–3; [21], p. 436).

A net {yα : α ∈ D} in a quasi-uniform space (Y, U) is called a right K-Cauchy net provided that, for each U ∈ U, there exists some α0 ∈ D such that

(yα, yβ) ∈ U for all α, β ∈ 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 ∈ A and U ∈ U, let

MA,U = {(f, g) ∈ C(X,Y ) × C(X,Y ):(f(x), g(x)) ∈ U for all x ∈ A}.

Then the collection {MA,U : A ∈ A and U ∈ U} 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 = {X}, UA is called the quasi-uniform topology of uniform convergence on C(X,Y ) and is denoted by UX . (2) If A = K(X)={K ⊆ X : K is compact}, UA is called the quasi-uniform topology of compact convergence on F (X,Y ) and is denoted by Uk. (3) If A = F (X)={A ⊆ X : A is ﬁnite}, 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 collection {MA,U : A ∈ A and U ∈ U} 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 {fα : α ∈ D} be a net in F (X,Y ) such that: (i) {fα : α ∈ D} 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 ∈ F (X,Y ) with f ∈ UX -cl(L(X,Y )). To show that f ∈ L(X,Y ), let x0 ∈ 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 ∈ U such that U[f(x0)] ⊆ H. By local symmetry, choose a symmetric V ∈ U such that 2 V [f(x0)] ⊆ U[f(x0)]. 2 Since U is a quasi-uniform space, choose W ∈ U such that W ⊆ V . Since f ∈ UX - cl(L(X,Y )) and MX,W [f] a is UX -neighborhood of f, there exists g ∈ L(X,Y ) such that g ∈ MX,W [f]. In particular, we have (f(y), g(y)) ∈ W ⊆ W 2 ⊆ V for all y ∈ X.

Since g is cl-supercontinuous at x0 and V [f(x0)] is an open neighborhood of f(x0) in Y , there exists a clopen set G containing x0 in X such that

g(y) ⊆ V [f(x0)] for all y ∈ G. Now let y ∈ G. Then −1 2 f(y) ∈ V [g(y)] = V [g(y)] ⊆ V [f(x0)] ⊆ U[f(x0)] ⊆ H.

Finally, UX -closedness of L(X,Y ) in C(X,Y ) follows from ([11], Lemma 2). 100 L. A. KHAN AND W. K. ALQURASHI

(b) Suppose Y is right K-complete, and let {fα : α ∈ I} be a right UX -K-Cauchy net in L(X,Y ). Let U ∈ U, and let x ∈ X be ﬁxed. There exists α0 ∈ I such that

(fα(y), fβ(y)) ∈ U for all α ≥ β ≥ α0 and y ∈ X; in particular, (fα(x), fβ(x)) ∈ U for all α ≥ β ≥ α0, and so {fα(x): α ∈ I} is a right K-Cauchy net in Y . Since Y is right K-complete, {fα(x)} is T (U)-convergent to a point f(x) ∈ Y . Hence we have f ∈ F (X,Y ) Up UX such that fα −→ f. Consequently, by Lemma 3.1, fα −→ f, and, by part (a), f ∈ L(X,Y ). Thus (L(X,Y ), UX ) is right K-complete. Corollary 3.3. ([8], Theorem 3.4) Let X be a topological space and (Y, U) a uniform space. Then (a) L(X,Y ) is UX -closed in both F (X,Y ) and C(X,Y ). (b) If Y is complete, then L(X,Y ) is UX -complete.

4. Up-Closedness and Compactness in C(X,Y)

We next consider Up-closedness/compactness of L(X,Y ) and other related sub- spaces of C(X,Y ). In this case, we also need to include extensions of several related results on joint continuity, equicontinuity and even continuity in the quasi-uniform setting. A topological space X is called sum connected ([9], p. 34) if each x ∈ X has a connected neighborhood or equivalently each component of X is open in X. Clearly, every connected space and every locally connected space is sum connected. Let X and Y be topological spaces. A function f : X → Y is said to be: (1) strongly continuous [12] if f(A) ⊆ f(A) for all A ⊆ X, or equivalently f(Ad) ⊆ f(A) for all A ⊆ X, where Ad denotes the set of all limit points of A. (2) perfectly continuous [17] if for every open set V ⊆ Y , f −1(V ) is clopen in X. Let P (X,Y ) (resp. S(X,Y )) denote the function space of all perfectly continuous (resp. strongly continuous) functions from X into Y . It is easy to see that S(X,Y ) ⊆ P (X,Y ) ⊆ L(X,Y ) ⊆ C(X,Y ). It is shown in ([12], p. 269) that if f ∈ S(X,Y ), then f(C) is a singleton for every non-empty connected subset C of X; conversely, if X locally connected, and f : X → Y a function such that f(C) is a singleton for every non-empty connected subset C of X, then f ∈ S(X,Y ). Further, in ([16], p. 167), it is shown that (1) if Y is Hausdorﬀ, then for any p f ∈ S(X,Y ) , f(C) is a singleton for every non-empty connected subset C of X; (2) if, in addition, X is locally connected, then S(X,Y ) is tp-closed in F (X,Y ). An extension of these results is obtained in ([9], p. 35) which we state for reference purpose as follows.

Theorem 4.1. Let f ∈ L(X,Y ) with Y a T0-space. Then (a) f(C) is a singleton for every nonempty connected subset C of X. p (b) If Y be a Hausdorﬀ space, then for any f ∈ L(X,Y ) , f(C) is a singleton for every non-empty connected subset C of X. (c) If X is sum connected and Y be a Hausdorﬀ space, then (i) L(X,Y ) = P (X,Y ) = S(X,Y ). (ii) L(X,Y ) = P (X,Y ) = S(X,Y ) is tp-closed in F (X,Y ). ARZELA-ASCOLI THEOREMS 101

Equicontinuity vs Even Continuity

Let X and Y be topological spaces. Then a family S ⊆ F (X,Y ) is said to be evenly continuous ([7], p. 235) if for each x ∈ X, each y ∈ Y and each neighborhood Hy of y, there is a neighborhood Gx of x and a neighborhood Jy of y such that f(x) ∈ Jy ⇒ f(Gx) ⊆ Hy. If Y = (Y, U) is a quasi-uniform space, a family S ⊆ F (X,Y ) is said to be equicontinuous at a point x0 ∈ X if for each U ∈ U, there exists a neighborhood G of x0 such that f(G) ⊆ U[f(x0)] for each f ∈ S. The family S is said to be equicontinuous if it is equicontinuous at every point of X [7, 2]. We shall require the following result which extends Theorems 7.22 and 7.23 of ([7], p. 226–227) from uniform to quasi-uniform spaces. Theorem 4.2. ([2], Proposition 3.3) Let X be a a topological space and (Y, U) a quasi-uniform space, and let S ⊆ C(X,Y ). (a) If S is equicontinuous, then S is evenly continuous. (b) If S is evenly continuous and if, for any ﬁxed point x0 ∈ X, S(x0) is relatively compact in Y , then S is equicontinuous at x0. p Equicontinuity of the closure S

It is well-known ([7], Theorem 7.14, p. 232) that if X is a Hausdorﬀ space, p (Y, U) a uniform space and S ⊆ C(X,Y ) is equicontinuous on X, then S is also equicontinuous on X. We do not know if this results holds for any class of quasi-uniform space. However, if we consider a weaker form of equicontinuity, then the above result holds for locally uniform spaces, as follows. Let X be a topological space, (Y, U) a quasi-uniform space, and S ⊆ F (X,Y ). We say that S is weakly equicontinuous ([6], p. 268; [18] ) if there exists an integer m such that for any x ∈ X and U ∈ U there exists a neighborhood Gx of x with m f(Gx) ⊆ U [f(x)] for all f ∈ S.

The following extends ([7], Theorem 7.14, p. 232) to locally uniform spaces. Lemma 4.3. Let X be a Hausdorﬀ space and (Y, U) a locally uniform space, p and let S ⊆ C(X,Y ). If S is weakly equicontinuous on X, then S is also weakly equicontinuous on X.

Proof. Let x0 ∈ X and U ∈ U be symmetric. We need to show that there exist an integer m and a neighborhood G = G(x0) of x0 such that m p f(G) ⊆ U [f(x0)] for all f ∈ S . (∗) p p Fix any f ∈ S , and let {fα : α ∈ I} be a net in S such that fα → f. Since S is weakly equicontinuous on X, there exists an integer n and a neighborhood G = G(x0) of x0 such that n g(G) ⊆ U [g(x0)] for all g ∈ S. 102 L. A. KHAN AND W. K. ALQURASHI

In particular, n fα(y) ∈ U [fα(x0)] for all y ∈ G and α ∈ I. p Fix any y ∈ G(x0). Since fα → f, choose α0 ∈ I such that (f(y), fα0 (y)) ∈ U and

(f(x0), fα0 (x0)) ∈ U. Hence, −1 n f(y) ∈ U [fα0 (y)] = U[fα0 (y)] ⊆ U[U [fα(x0)]] n n+2 ⊆ U[U [U[f(x0)]]] = U [f(x0)]. p This proves (∗) with m = n + 2. Thus S is weakly equicontinuous on X.

Joint Continuity of UX and Up topologies Let X and Y be topological spaces. A topology µ on F (X,Y ) is said to be jointly continuous (or admissible) on A ([7], p. 223) if the evaluation map e :(F (X,Y ), µ) × X → Y , defined by e(f, x) = f(x)(f ∈ C(X,Y ), x ∈ X), is continuous. A topology µ on F (X,Y ) is said to be jointly continuous on compacta if, for each compact K ⊆ X if the evaluation map e :(F (X,Y ), µ)×K → Y , defined by e(f, x) = f(x)(f ∈ C(X,Y ), x ∈ K), is continuous. It is well-known ([7], , p. 223) that if a topology µ on F (X,Y ) is jointly continuous on compacta, then tp ⊆ tk ⊆ µ; if (F (X,Y ), µ) is compact and Y is Hausdorff, then (F (X,Y ), p) is Hausdorff and consequently µ = tk = tp. If X is a locally compact regular space, then tk is the smallest jointly continuous topology on C(X,Y ) (in particular, tk is itself jointly continuous) (see ([7], p. 223; [4], Proposition 2.1, p.155). Consequently, the pointwise topology tp of C(X,Y ) is not jointly continuous if tp 6= k ([4], Cor), p. 156). In the case of (Y, U) a uniform space, it is shown in ([7], Theorem 7.9(b), p. 227) that topology UX on C(X,Y ) is jointly continuous. Its extension to quasi-uniform space setting has been given in ([16], Lemma 2.7, p. 769). We shall see below that the topology Up on an equicontinuous subset of C(X,Y ) is jointly continuous. Theorem 4.4. Let X be a topological space and (Y, U) a Hausdorff locally symmetric quasi-uniform space, and let S ⊆ C(X,Y ) be equicontinuous. Then (a) Up is a topology of joint continuity on S. (b) Up = Uk on S. Consequently, S is Up-compact iff it is Uk-compact.

Proof. (a) Let f ∈ F (X,Y ), x0 ∈ X, and U ∈ U . We need to show that there exists a Up-neighborhood N of f in F (X,Y ) and a neighborhood G of x0 in X such that g(y) ∈ U[f(x0)] for all g ∈ N ∩ S, y ∈ G.(∗∗) By local symmetry, choose a symmetric open V ∈ U be such that 2 V [f(x0)] ⊆ U[f(x0)].

Since S is equicontinuous, there exists an open neighborhood G of x0 such that

h(G) ⊆ V [h(x0)] for all h ∈ S.

Clearly N = N(x0,V [f(x0)]) = {g ∈ F (X,Y ): g(x0) ∈ V [f(x0)]} is a Up neighborhood of f in F (X,Y ). To vertify (∗∗), let g ∈ N ∩ S and y ∈ G. Then

g(y) ∈ V [g(x0)] ⊆ V [V [f(x0)]] ⊆ U[f(x0)].

Thus Up is a topology of joint continuity. ARZELA-ASCOLI THEOREMS 103

(b) Since each topology of joint continuity is ﬁner than the Uk, by (a) we have Uk ≤ Up; hence Up = Uk on S (since Up ≤ Uk in general). Corollary 4.5. ([5], Corollary 7, p. 251; [7], Theorem 7.15, p. 232) Let X be a topological space and (Y, U) a uniform space, and let S ⊆ C(X,Y ) be equicontinuous. Then S is Up-compact iﬀ it is Uk-compact.

Up-compactness of L(X,Y ) in C(X,Y )

Theorem 4.6. Let X be a sum connected and (Y, U) a Hausdorff quasi-uniform space. Then L(X,Y ) = P (X,Y ) = S(X,Y ) is equicontinuous and consequently also evenly continuous. Proof. Let x ∈ X, and let U ∈ U. (We need to show that there exists a neighborhood G of x such that f(G) ⊆ U[f(x)] for each f ∈ L(X,Y ).) Let Cx denote the component of X containing x. Since X is sum connected, Cx is open. In view of Theorem 4.1(a), for each f ∈ L(X,Y ), f(Cx) = {f(x)}. Clearly f(x) ∈ U[f(x)]. Taking G = Cx, we have f(G) ⊆ U[f(x)] for all f ∈ L(X,Y ). Thus L(X,Y ) is equicontinuous. By Theorem 4.2(a), L(X,Y ) is also evenly continuous. Theorem 4.7. (Arzela-Ascoli) Let X be a sum connected space and (Y, U) a Hausdorff locally symmetric quasi-uniform space. Then (a) Up = Uk on L(X,Y ). (b) If, in addition, X is a k-space and L(X,Y )(x) is compact in Y for each x ∈ X, then L(X,Y ) = P (X,Y ) = S(X,Y ) is Up-compact. Proof. (a) By Theorem 4.5, L(X,Y ) is equicontinuous. Then by Theorem 4.4(a), the topology Up on L(X,Y ) is jointly continuous and hence by Theorem 4.4(b), Up = Uk on L(X,Y ). (b) In this case, by Ascoli theorem of Kelley ([7], Theorem 7.18, p. 234, ), where Y is only assumed to be a Hausdorff topological space, it follows that L(X,Y ) is Uk-compact, hence, Up-compact by part (a). Remark 4.8. If (Y, U) is a Hausdorff uniform space, then from Theorems 4.6 and 4.7 we obtain ([9], Theorem 3.9 and Theorem 3.10) repectively, as corollaries.

Acknowledgments. The authors would like to thank Professors H. P. A. K¨unzi and R. A. McCoy for communicating to us useful information of various concepts used in this article and also the anonymous referee for his/her comments that helped us improve the presentation.

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Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah-21589, Saudi Arabia E-mail address: [email protected]

Department of Mathematics, Faculty of Science (Girls Section), King Abdulaziz University, P. O. Box 80203, Jeddah-21589, Saudi Arabia Current address: Department of Mathematical Sciences, College of Applied Sciences, Umm Al-Qura University, P. O. Box 7944, Makkah-24381, Saudi Arabia E-mail address: [email protected]