Set-open topologies on function spaces

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

Abstract

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.


Keywords

set-open topology; pseudocompact-open topology; C-compact- open topology; quasi-uniform convergence topology; right K- completeness; α-continuous function

Subject classification

54C35; 46A16; 54E15; 54C08.

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