Symmetry Conditions on the Coincidence of Some Notions of Quasi-Uniform Completeness

We generalize the notions of quietness and semisymmetry defined by Doitchinov (1991) and Deák (1991) and we study the role of these extended notions on the coincidence of some well-known quasi-uniform completeness. In particular, it is shown that the bicompletion coincides (up to quasi-uniformism) with the standard D-completion in quiet ⋆-weakly pair Cauchy bounded quasi-uniform spaces and it coincides with ⋆-half-com- pletion defined by Romaguera and Sánchez-Granero (2002), in T0-weakly quiet quasi-uniform spaces

Quasi-uniform hyperspaces of compact subsets

AbstractLet (X,u) be a quasi-uniform space, K(X) be the family of all nonempty compact subsets of (X,u). In this paper, the notion of compact symmetry for (X,u) is introduced, and relationships between the Bourbaki quasi-uniformity and the Vietoris topology on K(X) are examined. Furthermore we establish that for a compactly symmetric quasi-uniform space (X,u) the Bourbaki quasi-uniformity u∗ on K(X) is complete if and only if u is complete. This theorem generalizes the well-known Zenor-Morita theorem for uniformisable spaces to the quasi-uniform setting

Set-open topologies on function spaces

[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

Quasi-uniform convergence topologies on function spaces- Revisited

[EN] Let X and Y be topological space and F(X,Y) the set of all functions from X into Y. We study various quasi-uniform convergence topologies U_{A} (A⊆P(X)) on F(X,Y) and their comparison in the setting of Y a quasi-uniform space. Further, we study U_{A}-closedness and right K-completeness properties of certain subspaces of generalized continuous functions in F(X,Y) in the case of Y a locally symmetric quasi-uniform space or a locally uniform space.Alqurash, WK.; Khan, LA. (2017). Quasi-uniform convergence topologies on function spaces- Revisited. Applied General Topology. 18(2):301-316. doi:10.4995/agt.2017.7048SWORD30131618