Weakly developable and weakly k-developable spaces, and the Vietoris topology

Abstract

In this paper we investigate the notions of weakly developable and weakly k-developable space. We give a metrization theorem for weakly developable spaces, and characterize them by means of some generalized metric notions introduced in the paper which are weaker than those of wΔ-space or space having a Gδ∗-diagonal. We also introduce some notions stronger than the notion of p-sequence to characterize Tychonoff weakly developable and weakly k-developable spaces. These characterizations provide us with some applications and in particular we study hereditarity of weak developability and weak k-developability of the hyperspaces F(X) and K(X) endowed with the Vietoris (finite) topology. We prove that a Tychonoff space X is weakly developable if and only if F(X) is a weakly developable space. We show that K(X) for a Tychonoff weakly k-developable space X is weakly developable, and that K(X) for a Čech-complete space X which is developable or has a regular Gδ-diagonal is weakly k-developable.