() @ Appl. Gen. Topol. 18, no. 2 (2017), 301-316doi:10.4995/agt.2017.7048 c© AGT, UPV, 2017 Quasi-uniform convergence topologies on function spaces - Revisited Wafa Khalaf Alqurashi a and Liaqat Ali Khan b,∗ a Department of Mathematics, Faculty of Science (Girls Section), King Abdulaziz University, P.O. Box 80203, Jeddah-21589, Saudi Arabia. (wafa-math@hotmail.com) b Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah-21589, Saudi Arabia. (lkhan@kau.edu.sa) Communicated by H.-P. A. Künzi Abstract Let X and Y be topological spaces and F(X, Y ) the set of all functions from X into Y . We study various quasi-uniform convergence topologies UA (A ⊆ P(X)) on F(X, Y ) and their comparison in the setting of Y a quasi-uniform space. Further, we study UA-closedness and right K- completeness properties of certain subspaces of generalized continuous functions in F(X, Y ) in the case of Y a locally symmetric quasi-uniform space or a locally uniform space. 2010 MSC: 54C35; 54E15; 54C08. Keywords: quasi-uniform space; topology of quasi-uniform convergence on a family of sets; locally uniform spaces, right K-completeness; quasi-continuous functions; somewhat continuous functions. 1. Introduction 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 systematically studied by Kelley ([17], Chapter 7). It is shown there that: (i) Up ≤ Uk ≤ UX; (ii) C(X, Y ) ∗Corresponding author Received 24 December 2016 – Accepted 18 February 2017 http://dx.doi.org/10.4995/agt.2017.7048 W. K. Alqurashi and L. A. Khan is UX-closed in F(X, Y ); (iii) if Y is complete, then F(X, Y ) is UX-complete, hence C(X, Y ) is also UX-complete. Since every topological space is quasi-uniformizable ([8, 39]; [11], p. 27; [7], p. 34), we may assume that Y = (Y, U) with U a quasi-uniformity. Main advan- tage of this assumption is that one can introduce various notions of Cauchy nets and completeness. In this setting, some quasi-uniform convergence topologies UA (A ⊆ P(X)) on F(X, Y ) were first discussed by Naimpally [31]. In recent years, this topic has been further investigated by Papadopoulos [37, 38], Cao [6] and Kunzi and Romaguera [25, 26], among others. There is also a parallel notion of ”set-open topologies” SA (A ⊆ P(X)) on F(X, Y ) which were intro- duced by Fox [12] and further developed by Arens [2], Arens-Dugundji [3], and more recently in the papers [4, 9, 23, 33, 34, 35, 36]. These SA topologies are, in general, different from their corresponding uniform convergence topologies UA (A ⊆ P(X)) even in the case of Y a metric space, but the two notions coincide in some other particular cases. Regarding completeness in quasi-uniform spaces, the formulation of the no- tion of ”Cauchy net” or ”Cauchy filter” in such spaces has been fairly difficult, and has been approached by several authors (see, e.g., [1, 8, 10, 41, 42, 43, 44, 45]). We shall find it convenient to restrict ourselves to the notions of a ”right K-Cauchy net” and ”right K-complete space” on function spaces, as in [26]. In this paper, we consider various quasi-uniform convergence topologies on F(X, Y ) and study their comparison and equivalences. Further, we extend some results of above authors on closedness and completeness to more general classes of functions (not necessarily continuous). These include the subspaces of quasi-continuous, somewhat continuous and bounded functions [16, 22, 37, 40]. Here, we shall need to assume that Y is a locally symmetric quasi-uniform space or a locally uniform space (as appropriate), both notions being equivalent to Y a regular topological space [31, 46]. We have included multiple references for certain concepts for the convenience of readers to access the literature. Some open problems are also stated. 2. Preliminaries Definition 2.1 ([17], p. 175-176). Let Y be a non-empty set. For any U, V ⊆ Y × Y, we define U−1 = {(y, x) : (x, y) ∈ U} U ◦ V = {(x, y) ∈ Y × Y : ∃ z ∈ Y such that (x, z) ∈ U and (z, y) ∈ V }. If U = V , we shall write U◦U = U2. If U = U−1, then U is called symmetric. The subset ∆(Y ) = {(y, y) : y ∈ Y } of Y × Y is called the diagonal on Y . If △(Y ) ⊆ U, then clearly U ⊆ U ◦ U = U2 ⊆ U3 ⊆ ..... For any x ∈ Y , A ⊆ Y and U ⊆ Y × Y , let U[x] = {y ∈ Y : (x, y) ∈ U} and U[A] = ∪x∈AU[x]. Definition 2.2. A family U of subsets of Y × Y is called a quasi-uniformity on Y [7, 11, 24, 29] if it satisfies the following conditions: (QU1) △(Y ) ⊆ U for all U ∈ U. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 302 Quasi-uniform convergence topologies on function spaces - Revisited (QU2) If U ∈ U and U ⊆ V , then V ∈ U. (QU3) If U, V ∈ U, then U ∩ V ∈ U. (QU4) If U ∈ U, there is some V ∈ U such that V 2 ⊆ U. In this case, the pair (Y, U) is called a quasi-uniform space. If, in addition, U satisfies the symmetry condition: (U5) U ∈ U implies U −1 ∈ 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 [46] if U satisfies (QU1)- (QU3) and (U5). Definition 2.3. (i) A quasi-uniform space (Y, U) is called locally symmetric if, for each y ∈ Y and each U ∈ U, there is a symmetric V ∈ U such that V 2[y] ⊆ U[y] [11]. (ii) A semi-uniform space (Y, U) is called locally uniform [46] if, for each y ∈ Y and each U ∈ U, there is a V ∈ U such that V 2[y] ⊆ U[y]. Definition 2.4 ([11], p. 2-3; [46], p. 436). Let (Y, U) be 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 . Equivalently, for each y ∈ Y , the collection By = {U[y] : U ∈ U} forms a local base at y for the topology T (U). If (Y, τ) is a topological space, then a quasi-uniformity U on Y is said to be compatible with (Y, τ) provided τ = T (U). It is well-known that a topolog- ical space (Y, τ) is completely regular iff there exists a compatible uniformity U on Y . Csaszar [8] showed that every topological space has a compatible quasi-uniformity. In [39], Pervin greatly simplified Csaszar’s proof by giving a direct method of constructing a compatible quasi-uniformity for an arbitrary topological space. For more information, see ([29], p. 14-16; [7], p. 34). Definition 2.5. A net {yα : α ∈ D} in a topological space (Y, τ) is said to be τ-convergent to y ∈ Y if, for each τ-open neighborhood G of y in Y , there exists an α0 ∈ D such that yα ∈ G for all α ≥ α0 ([17], p. 65-66). In particular, a net {yα : α ∈ D} in a quasi-uniform or locally uniform space (Y, U) is said to be T (U)-convergent to y ∈ Y if, for each U ∈ U, there exists an α0 ∈ D such that yα ∈ U[y] for all α ≥ α0. Definition 2.6 ([41, 26, 24, 7]). Let (Y, U) be a quasi-uniform space. A net {yα : α ∈ D} in Y 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. [26], Lemma 1, p. 289). c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 303 W. K. Alqurashi and L. A. Khan Definition 2.7. Let (Y, U) be a quasi-uniform space or a locally uniform space, and let S ⊆ Y . Then: (i) S is called precompact [11, 29] if, given any U ∈ U, there exists a finite set F ⊆ Y such that S ⊆ U[F ]. (ii) S is called totally bounded [11, 29] if, given any U ∈ U, there exists a finite cover {G1, G2, ...., Gn} of S such that ∪ n i=1(Gi × Gi) ⊆ U. (iii) S is bounded [30] if given any U ∈ U, there exists an m ∈ N and a finite set F ⊆ Y , such that S ⊆ Um[F ] = ∪y∈F U m[y]. Note. By ([29], p. 49; [30], p. 368), for any S ⊆ (Y, U), a quasi-uniform space, S is totally bounded ⇒ S is precompact ⇒ S is bounded, but the converses need not be true ([11], p. 152; [29], p. 49). In fact, by [27], even a compact quasi-uniform space is not necessarily totally bounded. However, if (Y, U) is a uniform spaces, S is precompact iff S is totally bounded ([11], p. 52; [29], p. 49). If (X, τ) is a topological space and A ⊆ X, the closure of A is denoted by A τ or τ-cl(A) (or simply A or cl(A)); the interior of A is denoted by τ −int(A) (or simply int(A)). We shall denote the power set of X by P(X). 3. Quasi-uniform convergence topologies on F(X, Y ) 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(X) and U ∈ U, let MA,U = {(f, g) ∈ F(X, Y ) × F(X, Y ) : (f(x), g(x)) ∈ U for all x ∈ A}. Then the collection {MA,U : A ∈ A(X) and U ∈ U} forms a subbase for a quasi-uniformity, called the quasi-uniformity of quasi-uniform conver- gence on the sets in A(X) induced by U. The resultant topology on F(X, Y ) is called the topology of quasi-uniform convergence on the sets in A(X) and is denoted by UA [25, 26]. (i) If A = {X}, UA is called the quasi-uniform convergence topology on F(X, Y ) and is denoted by UX. (ii) If A = K(X)={A ⊆ X : A is compact}, UA is called the quasi- uniform compact convergence topology on F(X, Y ) and is de- noted by Uk (iii) If A = σK(X)={A ⊆ X : A is σ-compact}, UA is called the quasi- uniform σ-compact convergence topology on F(X, Y ) and is de- noted by Uσ. (iv) If A = σ0(X)={A ⊆ X : A is countable}, UA is called the quasi- uniform countable convergence topology on F(X, Y ) and is de- noted by Uσ0. (v) If A = K0(X)={A ⊆ X : A is finite}, UA is called the quasi-uniform pointwise convergence topology on F(X, Y ) and is denoted by Up. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 304 Quasi-uniform convergence topologies on function spaces - Revisited Since each of the collection A in (i)-(v) is closed under finite unions, the col- lection {MA,U : A ∈ A(X) and U ∈ U} actually forms a base for the topology UA (cf. [28], p. 7). Lemma 3.1. Let X be a topological space and (Y, U) a quasi-uniform space, and let A, B ⊆ X and U, V ∈ U be such that MA,U ⊆ MB,V . (i) If B 6= ∅, then U ⊆ V . (ii) If V 6= Y × Y, then B ⊆ A. Proof. (i) Suppose B 6= ∅, but U * V , and let (a, b) ∈ U with (a, b) /∈ V . Consider the constant functions f, g : X → Y defined by f(x) = a (x ∈ X), g(x) = b (x ∈ X). Then (f, g) ∈ MA,U, but (f, g) /∈ MB,V . Indeed, if A = ∅, then (f(A), g(A)) ∈ ∅ ⊆ U; if A 6= ∅ and x ∈ A, then (f(x), g(x)) = (a, b) ∈ U. Hence (f, g) ∈ MA,U. On the other hand, since B 6= ∅, for any x ∈ B, (f(x), g(x)) = (a, b) /∈ V , (f, g) /∈ MB,V . This contradicts MA,U ⊆ MB,V . (ii) Suppose B * A, and let x0 ∈ B\A. Since V 6= Y × Y , choose c, d ∈ Y such that (c, d) /∈ V . Fix (p, q) ∈ U. Define f, g : X → Y by f(x) = p if x ∈ A, f(x) = c if x ∈ X\A; g(x) = q if x ∈ A, g(x) = d if x ∈ X\A. Then (f, g) ∈ MA,U, but (f, g) /∈ MB,V . Indeed, if A = ∅, then (f(A), g(A)) ∈ ∅ ⊆ U; if A 6= ∅ and x ∈ A, then (f(x), g(x)) = (p, q) ∈ U. Hence (f, g) ∈ MA,U. On the other hand, since x0 ∈ B and (f(x0), g(x0)) = (c, d) /∈ V , (f, g) /∈ MB,V . This contradicts MA,U ⊆ MB,V . Therefore B ⊆ A. � Theorem 3.2. Let X be a Hausdorff topological space and (Y, U) a quasi- uniform space. Let Up, Uσ0, Uσ, Uk and UX be the topologies on F(X, Y ) as defined above. Then (a) Up ≤ Uk ≤ Uσ ≤ UX and Up ≤ Uσ0 ≤ Uσ. (b) Uk = UX iff X is compact. (c) Up = Uk iff every compact subset of X is finite. In particular, if X is discrete, then Up = Uk. (d) Uσ = UX iff X = A for some σ-compact subset A of X. (e) Uk = Uσ iff every σ-compact subset of X is relatively compact. (f) Uσ0 = UX iff X is separable. (g) Uσ0 ≤ Uk iff every countable subset of X is relatively compact. (h) Uσ0, Uσ and UX have the same bounded sets in F(X, Y ). Proof. (a) Clearly, K0(X) ⊆ K(X) ⊆ σK(X), and so Up ≤ Uk ≤ Uσ ≤ UX on F(X, Y ). Further, K0(X) ⊆ σ0(X) ⊆ σK(X), and so Up ≤ Uσ0 ≤ Uσ ≤ UX on F(X, Y ). (b) Suppose UX ≤ Uk, and let U ∈ U, with U 6= Y × Y . Then there exist a compact subset K of X and a V ∈ U such that MK,V ⊆ MX,U. By Lemma 3.1(ii), X ⊆ K = K (since X is Hausdorff). Thus X is compact. Conversely, suppose X is compact. To show UX ≤ Uk, take c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 305 W. K. Alqurashi and L. A. Khan arbitrary MX,V ∈ UX. Taking K = X, which is compact, MK,V ∈ Uk and MX,V ⊆ MK,V . Hence MX,V ∈ Uk, and so UX ≤ Uk. (c) Suppose Uk ≤ Up, and let K ⊆ X be a compact set and U ∈ U, with U 6= Y × Y . Then there exist a finite subset A of X and a V ∈ U such that MA,V ⊆ MK,U . By Lemma 3.1(ii), K ⊆ A = A; hence K is finite. Conversely, suppose that every compact subset of X is finite. To show Uk ≤ Up, take arbitrary MK,U ∈ Uk with K ⊆ X a compact set. Then K is finite. Taking A = K, MA,V ∈ Up and MA,U ⊆ MK,U. Hence MK,U ∈ Up, and so Uk ≤ Up . In particular, if X is discrete, then every compact subset of X is finite and hence Up = Uk. (d) Suppose that UX ≤ Uσ, and let U ∈ U, with U 6= Y × Y . Then there exist a σ-compact set A ⊆ X and a V ∈ U such that MA,V ⊆ MX,U. By Lemma 3.1(ii), X = A, as required. Conversely, suppose X = A for some σ-compact subset A of X. To show UX ≤ Uσ, take arbitrary MX,U ∈ UX. Clearly, MA,U ∈ Uσ and MX,U ⊆ MA,U ⊆ MA,U. Hence MX,U ∈ Uσ, and so UX ≤ Uσ. (e) Suppose that Uσ ≤ Uk and let A be any σ-compact subset of X. If U ∈ U, with U 6= Y × Y , then there exist a compact set B ⊆ X and a V ∈ U such that MB,V ⊆ MA,U. By Lemma 3.1(ii), A ⊆ B, which implies that A is also compact. Conversely, suppose that every σ- compact subset of X is relatively compact. Take arbitrary MA,U ∈ Uσ with A a σ-compact subset of X. Since A is compact, M A,U ∈ Uk and clearly, M A,U ⊆ MA,U. Hence MA,U ∈ Uk, and so Uσ ≤ Uk . (f) Suppose UX ≤ Uσ0. Then, for any U ∈ U, with U 6= Y ×Y , MX,U ∈ UX and hence MX,U ∈ Uσ0 . So there exist a countable set A ⊆ X and a V ∈ U such that MA,V ⊆ MX,U. By Lemma 3.1(ii), X = A and so X is separable. Conversely, suppose X is separable, and let A ⊆ X be countable set such that A = X. Take arbitrary MX,U ∈ UX. Clearly, MA,U ∈ Uσ0 and MX,U ⊆ MA,U ⊆ MA,U. Hence MX,U ∈ Uσ0, and so UX ≤ Uσ0. (g) Suppose that Uσ0 ≤ Uk and let A be any countable subset of X. If U ∈ U, with U 6= Y × Y , then there exist a compact set B ⊆ X and a V ∈ U such that MB,V ⊆ MA,U. By Lemma 3.1(ii), A ⊆ B, which implies that A is also compact. Conversely, suppose that every countable subset of X is relatively compact. To show Uσ0 ≤ Uk, take arbitrary MA,U ∈ Uσ0 with A a countable subset of X. Since A is compact, M A,U ∈ Uk and clearly, MA,U ⊆ MA,U. Hence MA,U ∈ Uk, and so Uσ0 ≤ Uk. (h) Since Uσ0 ≤ UX, every UX-bounded subset of F(X, Y ) is easily seen to be Uσ0-bounded. In fact, let S ⊆ F(X, Y ) be UX-bounded set. Then for arbitrary MA,U ∈ Uσ0 with A a countable subset of X and U ∈ U, there exists an m ∈ N and a finite set J ⊆ F(X, Y ) such c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 306 Quasi-uniform convergence topologies on function spaces - Revisited that S ⊆ (MX,U) m[J]. Then S ⊆ (MA,U) m[J], showing that S is Uσ0- bounded. On the other hand, suppose that there exists a Uσ0-bounded set T ⊆ F(X, Y ) which is not UX-bounded. Then there exist a V ∈ U such that T " (MX,V )n[K] = MX,V n[K] for all n ∈ N and all finite sets K ⊆ F(X, Y ). Choose sequences {hn} ⊆ T, {xn} ⊆ X such that (f(xn), hn(xn)) /∈ V n for all n ∈ N and all f ∈ F(X, Y ). Let A = {xn}. Then MA,V ∈ Uσ0, and hn /∈ (MA,V ) n[f] for any n ∈ N and f ∈ F(X, Y ); hence T " (MA,V )n[K] for any n ∈ N and finite set K ⊆ F(X, Y ). Therefore T is not Uσ0-bounded, a contradiction. � Now, let X be a completely regular Hausdorff space and Y = (E, τ) a Hausdorff topological vector space (TVS, in short) over K(= R or C) with a base WE(0) of balanced τ−neighborhoods of 0 in E ([18], Theorem 5.1), and let CB(X, E) denote the vector space of all continuous bounded functions from X into E. In this setting, the collection V = {VG : G ∈ WE(0)} is a uniformity on E, where VH = {(x, y) ∈ E × E : x − y ∈ H}. For any A ∈ A(X) and H ∈ WE(0), let M∗A,VH = {(f, g) ∈ CB(X, Y ) × CB(X, Y ) : (f(x), g(x)) ∈ VH for all x ∈ A}. Then the collection {M∗A,VH (0) : A ∈ A(X) and H ∈ WE(0)} forms a base of neighbourhood of 0 in CB(X, E) for a linear topology, denoted by tA. Indeed, this follows from ([18], Corollary 8.2) and the fact that M∗A,VH (0) = {g ∈ CB(X, Y ) : (0, g) ∈ M ∗ A,VH } = {g ∈ CB(X, Y ) : g(x) ∈ H for all x ∈ A} : = Ncb(A, H). The quasi-uniform topologies Up, Uσ0, Uσ, Uk and UX on CB(X, Y ) become the linear topologies, denoted by tp, tσ0, tσ, tk and tu in the terminology of [21]. Consequently, we can deduce the following from above two results: Corollary 3.3 ([21], Lemma 3.2). Let X be a completely regular Hausdorff space and (E, τ) a Hausdorff TVS. Suppose that A, B ⊆ X and that G, H ∈ WE(0) are such that Ncb(A, G) ⊆ Ncb(B, H). Then: (i) If B 6= ∅, then G ⊆ H. (ii) If W 6= E, then B ⊆ A. � Corollary 3.4 ([14, 15]; [21], Theorem 3.3). Let X be a completely regular Hausdorff space and (E, τ) a Hausdorff TVS. Then: (i) tσ = tu iff X = A for some σ-compact subset A of X. (ii) tk = tσ iff every σ-compact subset of X is relatively compact. (iii) tσ0 = tX iff X is separable. (iv) tσ0 ≤ tk iff every countable subset of X is relatively compact. (v) tσ0, tσ and tu have the same bounded sets in CB(X, Y ). � c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 307 W. K. Alqurashi and L. A. Khan Finally, in this section, we give a brief account of set-open topologies SA (A ⊆ P(X)) on F(X, Y ) and their comparison with the corresponding quasi- uniform convergence topologies UA (A ⊆ P(X)) Let X and Y be topological spaces and let A ⊆ P(X). For any A ∈ A and any open set H ⊆ Y , let N(A, H) = {f ∈ F(X, Y ) : f(A) ⊆ H}. Then the collection {N(A, H) : A ∈ A, open sets H ⊆ Y } form a subbase for a topology on F(X, Y ), called the set-open (or A-open ) topology generated by A and denoted by SA. In particular, if A = {X} (resp. K(X), σK(X), σ0(X), F(X)), then SA is called the uniform (resp. compact-open, σ-compact-open, countable-open, point-open) topology and denoted by Su (resp. Sk, Sσ, Sσ0, Sp). The relation between the set-open topology and the topology of uniform convergence on a family A ⊆ K(X) was investigated by Kelley ([17], p. 230) and McCoy and Ntantu ([28], p. 9) in the case of Y a uniform space (see also [23]). These SA topologies are, in general, different from their corresponding uniform convergence topologies UA even in the case of Y a metric space. More recently, there has been a renewed interest on the problem for coincidence of these two notions and some interesting partial answers have been obtained in [33, 34, 4, 35, 36]. 4. Closedness and completeness in function spaces The results of this section are motivated by those given in [17, 31, 26] regar- ding the closedness and completeness of C(X, Y ) and CA(X, Y ) in (F(X, Y ), UX). It is well-known (e.g., [17, 32]) that C(X, Y ) is UX-closed in F(X, Y ) but not necessarily Up-closed. Later, some authors obtained variants of these results for spaces of quasi-continuous, somewhat continuous and bounded functions in the case of Y a uniform space [16, 22, 37, 40]. In this section, we examine their UA-closedness and right K-completeness in the setting of Y a locally symmetric quasi-uniform or locally uniform spaces. Let X be a topological space and (Y, U) a quasi-uniform space. Let {fα : α ∈ D} be a net in F(X, Y ) and A ⊆ X. We recall from [25, 26] that: (i) {fα} is said to be right K-Cauchy in (F(X, Y ), UA) if, for any U ∈ U, there exists an index α0 ∈ D such that (fα, fβ) ∈ MA,U for all α ≥ β ≥ α0. (ii) {fα} is said to be UA-convergent to f ∈ F(X, Y ) if, for any U ∈ U, there exists an index α0 ∈ D such that (f, fα) ∈ MA,U for all α ≥ α0. In this case, we shall write fα UA −→ f. (iii) (F(X, Y ), UA) is called right K-complete if each right K-Cauchy net in (F(X, Y ), UA) is UA-convergent to some function in F(X, Y ). Lemma 4.1. Let X be a topological space and (Y, U) a quasi-uniform space, and let A ⊆ X. Let {fα : α ∈ D} be a net in F(X, Y ) such that (a) {fα : α ∈ D} is a right K-Cauchy net in (F(X, Y ), UA) (b) fα(x) → f(x) for each x ∈ A (i.e. fα Up −→ f on A). Then fα UA −→ f. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 308 Quasi-uniform convergence topologies on function spaces - Revisited Proof. Let U ∈ U. Choose V ∈ U such that V 2 ⊆ U. Since {fα} is right-K- Cauchy, there exists α0 ∈ D such that (fα, fβ) ∈ MA,U for all α ≥ β ≥ α0. We claim that (f, fγ) ∈ MA,U for all γ ≥ α0. Fix any γ ≥ α0 and x0 ∈ A. Since {fα(x0) : α ∈ D} −→ f(x0), also its subnet {fα(x0) : α ≥ γ}−→ f(x0). So there is α(x0) ∈ D with α(x0) ≥ γ such that (f(x0), fα(x0)(x0)) ∈ V . Since α(x0) ≥ γ ≥ α0, (f(x0), fα(x0)(x0)) ∈ V , (fα(x0)(x0), fγ(x0)) ∈ V ; hence (f(x0), fγ(x0)) ∈ V ◦ V ⊆ U. � The following result is due to Kunzi and Romaguera ([26], Proposition 1). Using Lemma 4.1, we can present a somewhat shorter proof of this result for reader’s benefit. Theorem 4.2. Let X be a topological space and Y = (Y, U) a right K-complete quasi-uniform space, and let A ⊆ P(X) which covers X. Then (F(X, Y ), UA ) is right K-complete. Proof. Let {fα : α ∈ D} be a right K-Cauchy net in (F(X, E), UA), and let U ∈ U and x ∈ X be fixed. Since A covers X, x ∈ Ax for Ax ∈ A. There exists α0 ∈ D such that for each α ≥ β ≥ α0, (fα, fβ) ∈ MAx,U for all α ≥ β ≥ α0. In particular, (fα(x), fβ(x)) ∈ U for all α ≥ β ≥ α0, and so {fα(x) : α ∈ D} is a right K-Cauchy net in Y. Since (Y, U) a right K-complete, {fα(x) : α ∈ D} is T (U)-convergent to a point f(x) ∈ Y. Hence we have an f ∈ F(X, Y ) such that fα Up −→ f. Consequently, by Lemma 4.1, fα UA −→ f. Thus (F(X, Y ), UA ) is complete. � We next obtain variants of some results in [26] for spaces of quasi-continuous, somewhat continuous and bounded functions in the setting of locally symmetric quasi-uniform and locally uniform spaces. A subset A of topological space X is called semi-open (or quasi-open) if there exists an open set G such that G ⊆ A ⊆ cl(G); equivalently, A is semi-open iff A ⊂ cl(int(A)). A function f : X → Y is said to be quasi- continuous [22, 40] if f−1(H) is semi-open in X for each open set H in Y , or equivalently, if, for each point x ∈ X and for each open set H ⊆ Y containing f(x), there exists a semi-open set A ⊆ X such that x ∈ A and f(A) ⊆ H. Let Q(X, Y ) denote the set of all quasi-continuous functions from X into Y . Theorem 4.3. Let X be a topological space and (Y, U) a locally symmetric quasi-uniform space, and let A ⊆ X. Let {fα : α ∈ D} be a net in Q(X, Y ) which is UA-convergent to f. Then f ∈ Q(X, Y ). Proof. Let x0 ∈ X and suppose H is any T (U)-open set containing f(x0) in Y . We need to show that there exists a semi-open set G containing x0 in X such that f(G ∩ A) ⊆ H. By definition of T (U), there exists a U ∈ U such that U[f(x0)] ⊆ H. By local symmetry, choose a symmetric V ∈ U such that c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 309 W. K. Alqurashi and L. A. Khan V 2[f(x0)] ⊆ U[f(x0)]. Next choose a closed W ∈ U such that W 2 ⊆ V Since fα UA −→ f, there exists α0 ∈ D such that (f, fα) ∈ MA,W for all α ≥ α0. In particular, we can write (f(z), fα0(z)) ∈ W ⊆ W 2 ⊆ V for all z ∈ X. Since fα0 is quasi-continuous at x0, there exists a semi-open set G containing x0 in X such that fα0(z) ⊆ V [f(x0)] for all z ∈ G ∩ A. Finally, let z ∈ G ∩ A. Then, using symmetry of V , we obtain f(z) ∈ V −1[fα0(z)] = V [fα0(z)] ⊆ V [V [f(x0)]] ⊆ U[f(x0)] ⊆ H. Therefore, f(G ∩ A) ⊆ H; hence f ∈ Q(X, Y ). � Theorem 4.4. Let X be a topological space and (Y, U) a locally symmetric quasi-uniform space. Then: (a) Q(X, Y ) is UX-closed in F(X, Y ). (b) If Y is right K-complete, then (Q(X, Y ), UX) is right K-complete. Proof. (a) This follows from Theorem 4.3. (b) Suppose Y is right K-complete. Let {fα : α ∈ D} be a right K-Cauchy net in (Q(X, Y ), UX). Let U ∈ U and let x0 ∈ X be fixed. Since {fα : α ∈ D} be a right K-Cauchy net, there exists α0 ∈ D such that (fα, fβ) ∈ MX,U for all α ≥ β ≥ α0. In particular (fα(x0), fβ(x0)) ∈ U for all α ≥ β ≥ α0. and so {fα(x0) : α ∈ D} is a right K-Cauchy net in Y . Since Y is right K-complete, {fα(x0)} is T (U)-convergent to a point f(x0) ∈ Y . Hence we have a function f ∈ F(X, Y ) such that fα Up −→ f. Consequently, by Lemma 4.1, fα UX −→ f, and, by part (a), f ∈ Q(X, Y ). Thus (Q(X, Y ), UX) is K- complete. � Corollary 4.5 ([22], Theorem 3.1; [40], Theorem 2.2). Let X be a topological space and (Y, U) a uniform space. Then: (a) Q(X, Y ) is UX-closed in F(X, Y ). (b) If Y is complete, then (Q(X, Y ), UX) is complete. A function f : X → Y is said to be somewhat continuous [13] if for each open set V in Y such that f−1(V ) 6= ∅, there exists a nonempty open set U in X such that U ⊂ f−1(V ); or equivalently, if, for any M ⊆ X, M is dense in X implies f(M) is dense in f(X) ([13], p. 6) . Let SW(X, Y ) denote the set of all somewhat continuous functions from X into Y . Theorem 4.6. Let X be a topological space and (Y, U) a locally symmetric quasi-uniform space. Let {fα : α ∈ D} be a net in SW(X, Y ) which is UX- convergent to f. Then f ∈ SW(X, Y ). c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 310 Quasi-uniform convergence topologies on function spaces - Revisited Proof. Let M be a dense subset of X. We need to show that f(M) is dense in f(X). Let y0 ∈ f(X), and let H be an open neighborhood of y0 in f(X). (We need to show that H ∩ f(M) 6= ∅.) There exists a G ∈ T (U) such that H = G ∩ f(X). Choose x0 ∈ X such that f(x0) = y0. Since G ∈ T (U), there exists U ∈ U such that U[y0] ⊆ G. By local symmetry, choose a symmetric V ∈ U such that V 2[y0] ⊆ U[y0]. Since (Y, U) is a quasi-uniform space, choose a closed W ∈ U such that W 2 ⊆ V . Since fα UX −→ f, there exists α0 ∈ D such that fα ∈ MX,W [f] for all γ ≥ α0. In particular, (f(z), fα0(z)) ∈ W ⊆ W 2 ⊆ V for all z ∈ X. Since fα0 is somewhat continuous, fα0(M) is dense in fα0(X). Since W [fα0(x0)] is a neighbohood of fα0(x0) in the T (U)-topology, there exists some m ∈ M such that fα0(m) ∈ W [fα0(x0)]. Then (f(x0), fα0(x0)) ∈ W and (fα0(x0), fα0(m)) ∈ W ; hence (f(x0), fα0(m)) ∈ W ◦ W ⊆ V . Since m ∈ M ⊆ X, (f(m), fα0(m)) ∈ V . Hence (f(x0), f(m)) ∈ V ◦ V −1 = V 2. Consequently, f(m) ∈ V 2[f(x0)] ⊆ U[f(x0)] = U[y0] ⊆ G ⊆ H. Thus f(m) ∈ H, and so H ∩f(M) 6= ∅. Hence f(M) is dense in f(X), showing that f ∈ SW(X, Y ). � Theorem 4.7. Let X be a topological space and (Y, U) a locally symmetric quasi-uniform space. Then: (a) SW(X, Y ) is UX-closed in F(X, Y ). (b) If Y is right K-complete, then (SW(X, Y ), UX) is right K-complete. Proof. (a) This follows from Theorem 4.6. (b) Suppose Y is right K-complete. Let {fα : α ∈ D} be a right K-Cauchy net in (SW(X, Y ), UX) . Let U ∈ U and let x0 ∈ X be fixed. There exists α0 ∈ D such that (fα, fβ) ∈ MX,U for all α ≥ β ≥ α0. In particular (fα(x0), fβ(x0)) ∈ U for all α ≥ β ≥ α0, and so {fα(x0) : α ∈ D} is a right K-Cauchy net in Y . Since Y is right K- complete, {fα(x0)} is T (U)-convergent to a point f(x0) ∈ Y . Hence we have a function f ∈ F(X, Y ) such that fα Up −→ f. Consequently, by Lemma 4.1, fα UX −→ f, and, by part (a), f ∈ SW(X, Y ). Thus (SW(X, Y ), UX) is right K-complete. � Corollary 4.8 ([16], Theorem 1, p. 32). Let X be a topological space and (Y, U) a uniform space. Then: (a) SW(X, Y ) is UX-closed in F(X, Y ). (b) If Y is complete, then (SW(X, Y ), UX) is complete. We next present analogues of some results from [37] for functions having range as precompact or bounded sets. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 311 W. K. Alqurashi and L. A. Khan Theorem 4.9. Let X be a non-empty set and (Y, U) a locally uniform space. Let A ⊆ P(X) and BA(X, Y ) ⊆ F(X, Y ) the set of all functions which are bounded on each member of A. Let {fα : α ∈ D} be a net in BA(X, Y ) which is UA-convergent to f. Then f ∈ BA(X, Y ). (Hence BA(X, Y ) is UA-closed in F(X, Y ).) Proof. Let U ∈ U be symmetric and A ∈ A(X). Choose an α0 ∈ D such that (f, fα0) ∈ MA,U for all α ≥ α0. In particular, this implies that for each x ∈ A, f(x) ∈ U−1[fα0(x))] = U[fα0(x))] ⊆ U[fα0(A))]; hence f(A) ⊆ U[fα0(A)]. But fα0 ∈ BA(X, Y ), so there exists an integer m ≥ 1 and a finite set F ⊆ Y , such that fα0(A) ⊆ U m[F ]. Thus f(A) ⊆ U[fα0(A)] ⊆ U[U m[F ] = Um+1[F ], which means that f ∈ B(X, Y ). � Problem 4.10. The authors do not know whether or not the above result can be established for (Y, U) a locally symmetric quasi-uniform space. Theorem 4.11. Let X be a non-empty set and (Y, U) a uniform space, and let A ⊆ P(X) and PCA(X, Y ) ⊆ F(X, Y ) be the set of all functions which have precompact range on each member of A. Let {fα : α ∈ D} be a net in PCA(X, Y ) which is UA-convergent to f. Then f ∈ PCA(X, Y ). (Hence PCA(X, Y ) is UA-closed in F(X, Y ) and is UA-complete if (Y, U) is complete.) Proof. Let U ∈ U and A ∈ A(X). Choose symmetric W ∈ U with W ◦W ⊆ U. There exists a α0 ∈ D such that for each α ≥ α0, fα ∈ MA,W [f]. In particular, (f, fα0) ∈ MA,W , which implies that f(A) ⊆ W −1[fα0(A)] = W [fα0(A)]. But fα0 ∈ PCA(X, Y ), so there exists a finite set F ⊆ Y , such that fα0(A) ⊆ W [F ]. Thus f(A) ⊆ (W ◦ W)[F ] ⊆ U[F ], which means that f ∈ PCA(X, Y ). � Problem 4.12. The authors do not know whether or not the above result can be established for (Y, U) a locally symmetric quasi-uniform space or a locally uniform space. Next we consider the notion of functions f ∈ F(X, Y ) which are ”small off compact set”. First, let Y = E, a TVS over K (=R or C) with WE(0) a base of balanced neighborhoods of 0 in E. Recall that: a function f : X → E is called small off compact set (or vanish at infinity) [5, 18, 20] if, for each G ∈ WE(0), there exists a compact set K ⊆ X such that f(x) ∈ G for all x ∈ X\K. Note that if f ∈ F(X, E) is small off compact set, then given any G ∈ WE(0), there exists a compact set K ⊆ X such that f(x) − f(y) ∈ G for all x, y ∈ X\K. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 312 Quasi-uniform convergence topologies on function spaces - Revisited In fact, choose a balanced (or symmetric) H ∈ WE(0) such that H + H ⊆ G. Since f ∈ F0(X, E), there exists a compact set K ⊆ X such that f(x) ∈ H for all x ∈ X\K. Then, for any x, y ∈ X\K, f(x) − f(y) ∈ H − H = H + H ⊆ G. Motivated by this observation, we can introduce the notion of ”small off compact set” in the setting of quasi-uniform spaces, as follows. Let X be a topological space and (Y, U) a quasi-uniform space. (i) A function f : X → Y is said to be small off compact set if, for each U ∈ U, there exists a compact set K ⊆ X such that (f(x), f(y)) ∈ U for all x, y ∈ X\K. (ii) f : X → Y is said to have compact support if there exists a compact set K ⊆ X such that (f(x), f(y)) ∈ U for all x, y ∈ X\K and all U ∈ U. Note that a quasi-uniform space (Y, U) is T1 iff ∩{U : U ∈ U} = ∆(Y ) holds ([11], p. 6; [29], p. 36). Hence, in this case, the condition in (ii) is equivalent to: f(x) = f(y) for all x, y ∈ X\K. Let F0(X, Y ) (resp. F00(X, Y )) denote the subset of F(X, Y ) consisting of those functions which are small of compact set (resp. have compact support), and let C0(X, Y ) = F0(X, Y )∩C(X, Y ) and C00(X, Y ) = F00(X, Y )∩C(X, Y ). Clearly, F00(X, Y ) ⊆ F0(X, Y ) and C00(X, Y ) ⊆ C0(X, Y ). Lemma 4.13. C0(X, Y ) ⊆ B(X, Y ). Proof. Let f ∈ C0(X, Y ), and let U ∈ U. There exists a compact set K ⊆ X such that (f(x), f(y)) ∈ U for all x, y ∈ X\K. In particular, for a fixed x0 ∈ X\K, f(y) ∈ U[f(x0)] for all y ∈ X\K. Since f(K) is compact and hence bounded in Y ([30], p. 368), there exists an integer m and a finite set F ⊆ Y , such that f(K) ⊆ Um[F ]. Taking F1 = F ∪{f(x0)}, we have f(X) ⊆ U m[F1], which means that f ∈ B(X, Y ). � Theorem 4.14. Let X be a topological space and (Y, U) a uniform space. Then both F0(X, Y ) and C0(X, Y ) are UX-closed in F(X, Y ). Proof. Let f ∈ F(X, Y ) with f ∈ UX − cl(F0(X, Y )). Let U ∈ U. Choose a symmetric V ∈ U such that V 3 ⊆ U. There exists g ∈ F0(X, Y ) such that (f(x), g(x)) ∈ V for all x ∈ X. There exists a compact set K ⊆ X such that (g(x), g(y)) ∈ V for all x, y ∈ X\K. Then, for any x, y ∈ X\K, (f(x), g(x)) ∈ V, (g(x), g(y)) ∈ V, (g(y), f(y)) ∈ V −1. Hence (f(x), f(y) ∈ V ◦ V ◦ V −1 = V ◦ V ◦ V ⊆ U for all x, y ∈ X\K. Therefore f ∈ F0(X, Y ), and so F0(X, Y ) is UX-closed in F(X, Y ). By ([17], Theorem 7.9), C(X, Y ) is UX-closed in F(X, Y ). Thus C0(X, Y ) is also UX- closed in F(X, Y ). � c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 313 W. K. Alqurashi and L. A. Khan Remark 4.15. If X is not locally compact, then C0(X, Y ) may consist of only constant functions. For example, if X = Q (rationals), then C0(X, R) = {0} ([20], p. 12). Problem 4.16. If X is locally compact and Y = E, a topological vector space, then it is well-known that C00(X, E) is Uk-dense in C(X, E) and also UX- dense in C0(X, E) ([5], p. 96-98; [18], p. 81; [19], Theorem 3.2; [20], Theorem 1.1.10). We leave an open problem that whether or not these denseness results hold for Y a uniform space. Acknowledgements. The authors wish to thank Professors H. P. A. Künzi, R. A. McCoy and A. V. Osipov for communicating to us useful information of some concepts used in this paper, and the referee for suggesting a number of improvements to the original version of the paper. References [1] A. Andrikopoulos, Completeness in quasi-uniform spaces, Acta Math. Hungar. 105 (2004), 151–173. [2] R. F. Arens, A topology for spaces of transformations, Ann. of Math. 47, no. 2 (1946), 480–495. [3] R. Arens and J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951), 5–31. [4] A. Bouchair and S. Kelaiaia, Comparison of some set open topologies on C(X,Y), Topol- ogy Appl. 178 (2014), 352–359. [5] R. C. Buck, Bounded continuous functions on a locally compact space, Michigan Math. J. 5 (1958), 95–104. [6] J. Cao, Answers to two questions of Papadopoulos, Questions Answers Gen. Topology 14 (1996), 111–116. [7] S. Cobzas, Functional Analysis in Asymmetric Normed Spaces, Springer, 2013. [8] A. Csaszar, Fondements de la topologie generale, Akademiai Kiado, Budapest, 1960; Translation, Foundations of General Topology, Macmillan Company, 1963. [9] A. Di Concilio and S.A. Naimpally, Some proximal set-open topologies, Boll. Unione Mat. Ital. (8) 1-B (2000), 173–191. [10] D. Doitchinov, A concept of completeness of quasi-uniform spaces, Topology Appl. 38 (1991), 205–217. [11] P. Fletcher and W.F. Lindgren, Quasi-uniform spaces, Lecture Notes in Pure and Ap- plied Mathematics, 77, Marcel Dekker, Inc., 1982. [12] R. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51 (1945), 429–432. [13] K. R. Gentry, H. B. Hoyle, III, Somewhat continuous functions, Czechoslovak Math. J. 21 (1971), 5–12. [14] D. Gulick, The σ-compact-open topology and its relatives, Math. Scand. 30 (1972) 159– 176. [15] D. Gulick and J. Schmets, Separability and semi-norm separability for spaces of bounded continuous functions, Bull. Soc. Roy. Sci. Liège 41 (1972), 254–260. [16] H. B. Hoyle, III, Function spaces for somewhat continuous functions, Czechoslovak Math. J. 21 (1971), 31–34. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 314 Quasi-uniform convergence topologies on function spaces - Revisited [17] J. L. Kelley, General Topology, D. Van Nostrand Company, New York, 1955. [18] J. L. Kelley, I. Namioka and Co-authors, Linear Topological Spaces, D. Van Nostrand, 1963. [19] L. A. Khan, Some approximation results for the compact-open topology, Periodica Math. Hungar. 30 (1995), 81–86. [20] L. A. Khan, Linear Topological Spaces of Continuous Vector-valued Functions, Academic Publications, 2013 (DOI:0.12732/acadpubl.201301). [21] L. A. Khan and K. Rowlands, The σ-compact-open topology and its relatives on a space of vector-valued functions, Boll. Unione Mat. Italiana (7) 5-B (1991), 727–739. [22] J. L. Kohli and J. Aggarwal, Closedness of certain classes of functions in the topology of uniform convergence, Demonstratio Math. 45(2012), 947-952. [23] S. Kundu and R. A. McCoy, Topologies between compact and uniform convergence on function spaces, Internat. J. Math. Math. Sci. 16, no. 1 (1993), 101–110. [24] H. P. A. Künzi, An introduction to quasi-uniform spaces, in: Beyond topology, Contemp. Math., 486, Amer. Math. Soc., Providence, RI, 2009, pp. 239–304 [25] H. P. A. Künzi and S. Romaguera, Completeness of the quasi-uniformity of quasi-uniform convergence, Papers on general topology and applications (Gorham, ME, 1995), 231-237, Ann. New York Acad. Sci., 806, New York Acad. Sci., New York, 1996. [26] H. P. A. Künzi and S. Romaguera, Spaces of continuous functions and quasi-uniform convergence, Acta Math. Hungar. 75 (1997), 287–298. [27] P. Th. Lambrinos, On precompact (quasi-) uniform structures, Proc. Amer. Math. Soc. 62 (1977), 365–366. [28] R. A. McCoy and I. Ntantu, Topological Properties of Function Spaces, Lectures Notes in Math. No. 1315, Springer-Verlag, 1988. [29] M. G. Murdeshwar, and S. A. Naimpally, Quasi-uniform Topological Spaces, Noordhoff Groningen, 1966. [30] M. G. Murdeshwar and K. K. Theckedath, Boundedness in a quasi-uniform space, Canad. Math. Bull. 13 (1970), 367–370. [31] S. A. Naimpally, Function spaces of quasi-uniform spaces, Indag. Math. 27 (1966), 768– 771. [32] S. A. Naimpally, On strongly continuous functions, Amer. Math. Monthly 74 (1967), 166–169. [33] S. E. Nokhrin, Some properties of set-open topologies, J. Math. Sci. 144 (2007), 4123– 4151. [34] S. E. Nokhrin and A.V. Osipov, On the coincidence of set-open and uniform topologies, Proc. Steklov Inst. Math. 267, Suppl. 3 (2009), 184–191. [35] A. V. Osipov, The set-open topology, Topology Proc. 37 (2010), 205–217. [36] A. V. Osipov, Uniformity of uniform convergence on the family of sets, Topology Proc. 50 (2017), 79–86. [37] B. Papadopoulos, (Quasi) Uniformities on the set of bounded maps,Internat. J. Math. & Math. Sci. 17 (1994), 693–696. [38] B. Papadopoulos, A note on the paper ”Quasi-uniform convergence on function spaces”, Q & A in General Topology 13 (1995), 55–56. [39] W. J. Pervin, Quasi-uniformization of topological spaces, Math. Ann. 147 (1962), 316– 317. [40] M. Racilă, On some properties of quasi-uniform convergence and quasi-continuous func- tions, Real Anal. Exchange 24 (1998/99), 791–797. [41] S. Romaguera, On hereditary precompactness and completeness in quasi-uniform spaces, Acta Math. Hungar. 73 (1996), 159–178. [42] J. L. Sieber and W.J. Pervin, Completeness in quasi-uniform spaces, Math. Ann. 158 (1965), 79–81. [43] M. B. Smyth, Completeness of quasi-uniform and syntopological spaces, J. London Math. Soc. 49, no. 2 (1994), 385–400. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 315 W. K. Alqurashi and L. A. Khan [44] R. Stoltenberg, Some properties of quasi-uniform spaces, Proc. London Math. Soc. 17, no. 3 (1967), 226–240. [45] Ph. Sünderhauf, Quasi-uniform completeness in terms of Cauchy nets, Acta Math. Hun- gar. 69 (1995), 47–54. [46] J. Williams, Locally uniform spaces, Trans. Amer. Math. Soc. 168 (1972), 435–469. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 316