Set-open topologies on function spaces

Abstract

[EN] Let X and Y be 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). We study various set-open topologies tλ (λ ⊆ P(X)) on F(X,Y) and consider their existence, comparison and coincidence in the setting of Y a general topological space as well as for Y = R. Further, we consider the parallel notion of quasi-uniform convergence topologies Uλ (λ ⊆ P(X)) on F(X,Y) to discuss Uλ-closedness and right Uλ-K-completeness properties of a certain subspace of F(X,Y) in the case of Y a locally symmetric quasi-uniform space. We include some counter-examples to justify our comments.The authors wish to thank Professors H. P. A. K ̈unziand R. A. McCoy for communicating to us useful information of various con-cepts used in this paper and also the anonymous referee for his/her commentsthat helped us to correct some errors and improve the presentation.Alqurashi, WK.; Khan, LA.; Osipov, AV. (2018). Set-open topologies on function spaces. Applied General Topology. 19(1):55-64. doi:10.4995/agt.2018.7630SWORD556419