Set-open topologies on function spaces

Abstract

<p>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.</p>

Lightly chaotic functional envelopes

“…The first concerning the introduction of other light dynamical properties and the study of their interdependencies with classical dynamical properties, the second one concerning their connections with dynamical properties of the functional envelope. It might be worth considering set-open topologies and uniform convergence topologies (see for example [4], [13], [16], [17], [18]). Moreover, analogously, it might be interesting to study the connections when the envelope is an hyperspace.…”

“…Several topologies can be defined on the set Y X = {f : X → Y }, where X, Y are topological spaces (see [3], [4], [13], [16], [17], [18]). Among them we shall consider the compact-open topology, the point-open topology and the uniform convergence topology.…”

On homeomorphism groups and the set-open topology

“…This topology was first introduced by Arens and Dugundji [3]. Now the set-open topology effective is used in investigation of the topological properties of function spaces and the groups of all self-homeomorphisms of topological spaces [1,5,8,9,15,16,17,18,19,21]. We will denote as H B (X) and H B (X) the set of all homeomorphisms of X equipped with the B-open topology and B-open topology respectively.…”