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

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.

### MSC:

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