On the infimum of proximal topologies of the hyperspace

Abstract

We investigate properties of a topology ${\tau _{\delta \left( {\inf } \right)}},$ which is the infimum of a family ${\mathcal{T}_\delta }$ of all ρ-proximal topologies ${\tau _{\delta \left( \rho  \right)}}$ on the hyperspace of a metrisable topological space. As a main result a necessary and sufficient condition of a coincidence $\inf {\mathcal{T}_\delta } = {\tau _{\delta \left( \rho  \right)}}$ (i. e. the condition of existing the minimum) is found. Besides the relations between the topology ${\tau _{\delta \left( {\inf } \right)}}$ and the infimum of a family ${\mathcal{T}_W}$ of all Wijsman topologies, the infimum of a family ${\mathcal{T}_H}$ of all Hausdorff metric topologies, and the Fell topology are studied. In the second part of the article we consider a continuous map $X\xrightarrow{f}Y$ and its extension ${\exp _{\delta \left( {\inf } \right)}}X\xrightarrow{{\overline f }}{\exp _{\delta \left( {\inf } \right)}}Y$ on the hyperspaces$\exp X$, $\exp Y$, endowed with the topology ${\tau _{\delta \left( {\inf } \right)}}$, where $\overline f (F) = {\left[ {f(F)} \right]_Y}$ $({\left[  \cdot  \right]_Y}$ is a closure operator in the space Y). For the map $\overline f $ we’ve got a sufficient condition of its continuity. This condition is also necessary, if Y is a locally compact space with countable base.