Set-open topologies on function spaces

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

doi:10.4995/agt.2018.7630

Authors

Wafa Khalaf Alqurashi Umm Al-Qura University
Liaqat Ali Khan Quaid-i-Azam University https://orcid.org/0000-0002-0263-7270
Alexander V. Osipov Ural State University of Economics

DOI:

https://doi.org/10.4995/agt.2018.7630

Keywords:

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

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 âŠ† 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 âŠ† 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.

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Author Biographies

Wafa Khalaf Alqurashi, Umm Al-Qura University

Department of Mathematical Science, College of Applied Sciences

Liaqat Ali Khan, Quaid-i-Azam University

Department of Mathematics

Alexander V. Osipov, Ural State University of Economics

Krasovskii Institute of Mathematics and Mechanics, Ural Federal University

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