Hyperspace Selections Avoiding Points

TL;DR

This paper explores hyperspace selection problems in connected spaces, providing characterizations of compact orderable and weakly cyclically orderable spaces through selection properties and ternary relations.

Contribution

It introduces new characterizations of compact orderable and weakly cyclically orderable spaces based on hyperspace selection properties.

Findings

Characterization of compact orderable spaces via selection properties.

Equivalence of selection properties for two-point sets to the existence of cyclic orders.

Introduction of weakly cyclically orderable spaces.

Abstract

In this paper, we deal with a hyperspace selection problem in the setting of connected spaces. We present two solutions of this problem illustrating the difference between selections for the nonempty closed sets, and those for the at most two-point sets. In the first case, we obtain a characterisation of compact orderable spaces. In the latter case -- that of selections for at most two-point sets, the same selection property is equivalent to the existence of a ternary relation on the space, known as a cyclic order, and gives a characterisation of the so called weakly cyclically orderable spaces.