The Vietoris hyperspace of finite sets of Erd\H{o}s space

TL;DR

This paper proves that the Vietoris hyperspace of finite sets of Erdős space is homeomorphic to Erdős space itself, revealing a self-similarity property of this topological space.

Contribution

It establishes that the hyperspace of finite subsets of Erdős space is topologically identical to Erdős space, extending previous results about Erdős space factors.

Findings

Vietoris hyperspace of finite sets of Erdős space is homeomorphic to Erdős space.

Supports the idea that Erdős space exhibits self-similarity in its hyperspaces.

Enhances understanding of the topological structure of Erdős space and its hyperspaces.

Abstract

Recently, David S. Lipham has shown that if $X$ is an Erd\H{o}s space factor then the Vietoris hyperspace $\mathcal{F}(X)$ of finite subsets of $X$ is an Erd\H{o}s space factor. In this short note we prove that if $\mathfrak{E}$ denotes Erd\H{o}s space then $\mathcal{F}(\mathfrak{E})$ is in fact homeomorphic to $\mathfrak{E}$.