Set-open topologies on function spaces. (English) Zbl 1392.54020

Alqurashi, Wafa Khalaf; Khan, Liaqat Ali; Osipov, Alexander V.

Appl. Gen. Topol. 19, No. 1, 55-64 (2018).

Let \(X\) and \(Y\) be topological spaces and \(F(X,Y)\) (\(C(X,Y)\)) be the set of all (continuous) functions \(f:X\to Y\). The authors consider different types of topologies defined on \(F(X,Y)\). The first type consists of various set-open topologies defined as follows. Let \(\lambda\subset\mathcal{P}(X)\) be a network which covers \(X\). For any \(A\in\lambda\) and open \(G\subset Y\), let \(N(A,G) = \{f \in F(X,Y) : f(A)\subset G\}\). Then the collection \(\{N(A,G) : A\in\lambda, \text{ open } G \subseteq Y \}\) forms a subbase for a “set-open” topology on \(F(X, Y)\), called the \(\lambda\)-open topology and denoted \(t_\lambda\). In the first part of the paper under review the authors study existence, comparison and coincidence of such topologies in the setting of \(Y\) a general topological space as well as for \(Y = \mathbb{R}\). In addition to the \(t_\lambda\)-topology, the authors define the notion of \(t_{\lambda^\ast}\)-topology on \(C(X,Y)\) which has as a subbase the collection \(\{ N^\ast_c(A,G): A\in\lambda, \text{ open }G\subset Y\}\), where \(N^\ast_c(A,G)=\{ f\in C(X,Y): \overline{f(A)}\subset G\}\).
In the next parts of the paper the authors also consider the topology of uniform convergence on elements of \(\lambda\) on \(C(X,Y)\), denoted by \(C_{\lambda,u}(X,Y)\), which has as a base at each \(f\in C(X,Y)\) the collection \(\{ \langle f,A, \varepsilon\rangle: A\in\lambda, \varepsilon> 0\}\), where \(\{ \langle f,A, \varepsilon\rangle=\{ g\in C(X,Y): \operatorname{sup}_{x\in A}\varrho(f(x),g(x))<\varepsilon\}\).
Finally the authors consider the notions of quasi-uniform convergence topologies on \(F(X,Y)\), which are parallel to those of the set-open topologies. Let \((Y, \mathcal{U})\) be a quasi-uniform space, and let \(\lambda\subset\mathcal{P}(X)\) be a collection which covers \(X\). For any \(A\in\lambda\) and \(U\in\mathcal{U}\), let \(\hat{U}|A=\{ (f,g)\in F(X,Y)\times F(X,Y): (f(x),g(x))\in U \text{ for all }x\in A\}\). Then the collection \(\{ \hat{U}|A : A\in\lambda \text{ and }U\in \mathcal U\}\) forms a subbase for a quasi-uniformity, called the quasi-uniformity of quasi-uniform convergence on the sets in \(\lambda\) induced by \(\mathcal{U}\). The induced topology on \(F(X, Y)\) is called the topology of quasi-uniform convergence on the sets in \(\lambda\) and is denoted by \(\mathcal{U}_\lambda\). The authors establish some results on closedness and completeness of the space \(C^\alpha(X, Y)\) of all \(\alpha\)-continuous functions from \(X\) into \(Y\) in the \(\mathcal{U}_X\)-topology.

Reviewer: Tomasz Natkaniec (Gdańsk)

MSC:

54C35	Function spaces in general topology
54C30	Real-valued functions in general topology
54E15	Uniform structures and generalizations
54C08	Weak and generalized continuity

Keywords:

set-open topology; pseudocompact-open topology; \(C\)-compact-open topology; quasi-uniform convergence topology; right \(K\)-completeness; \(\alpha\)-continuous function

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