A note on hyperspaces by closed sets with Vietoris topology

TL;DR

This paper studies the properties of hyperspaces of non-empty closed sets with the Vietoris topology when the base space is an infinite countable discrete space, revealing embeddings, tightness relations, and conditions for generalized metric properties.

Contribution

It provides new insights into the structure and properties of hyperspaces over countable discrete spaces, including embeddings, tightness, and characterizations of gamma-spaces.

Findings

Hyperspace contains a closed copy of n-th power of Sorgenfrey line.

Tightness of hyperspace equals the set-tightness of the base space.

Characterization of when hyperspace is a gamma-space.

Abstract

For a topological space $X$, let $CL(X)$ be the set of all non-empty closed subset of $X$, and denote the set $CL(X)$ with the Vietoris topology by $(CL(X), \mathbb{V})$. In this paper, we mainly discuss the hyperspace $(CL(X), \mathbb{V})$ when $X$ is an infinite countable discrete space. As an application, we first prove that the hyperspace with the Vietoris topology on an infinite countable discrete space contains a closed copy of $n$-th power of Sorgenfrey line for each $n\in\mathbb{N}$. Then we investigate the tightness of the hyperspace $(CL(X), \mathbb{V})$, and prove that the tightness of $(CL(X), \mathbb{V})$ is equal to the set-tightness of $X$. Moreover, we extend some results about the generalized metric properties on the hyperspace $(CL(X), \mathbb{V})$. Finally, we give a characterization of $X$ such that $(CL(X), \mathbb{V})$ is a $\gamma$-space.