The compact strong Z-set property in a hyperspace of finite subsets

TL;DR

This paper proves that in a certain hyperspace of finite subsets of a connected, locally path-connected space, every compact set is a strong Z-set, revealing a significant topological property of these hyperspaces.

Contribution

It establishes that all compact sets in the hyperspace Fin(X) are strong Z-sets, extending understanding of the topological structure of hyperspaces of finite subsets.

Findings

Every compact set in Fin(X) is a strong Z-set.

The result applies to non-degenerate, connected, locally path-connected metrizable spaces.

Enhances the theory of hyperspaces and their topological properties.

Abstract

Let X be a non-degenerate, connected, locally path-connected metrizable space and Fin(X) be the hyperspace consisting of non-empty finite subsets in X endowed with the Vietoris topology. In this paper, we show that every compact set in Fin(X) is a strong Z-set.