Hausdorff hyperspaces of $R^m$ and their dense subspaces

TL;DR

This paper explores the topological structure of hyperspaces of closed sets in Euclidean spaces and the real line, showing certain dense subspaces are homeomorphic to Hilbert spaces, revealing their infinite-dimensional topology.

Contribution

It demonstrates that various dense subspaces of hyperspaces in $\mathbb{R}^m$ are homeomorphic to Hilbert spaces, and characterizes components of hyperspaces in $\mathbb{R}$.

Findings

Dense subspaces of $CLB_H(\mathbb{R}^m)$ are homeomorphic to $\ell_2$.

Certain components of $CL_H(\mathbb{R})$ are homeomorphic to $\ell_2(2^{\aleph_0})$.

Nonseparable components avoiding specific sets are Hilbert spaces.

Abstract

Let $CLB_H(X)$ denote the hyperspace of closed bounded subsets of a metric space $X$, endowed with the Hausdorff metric topology. We prove, among others, that natural dense subspaces of $CLB_H(R^m)$ of all nowhere dense closed sets, of all perfect sets, of all Cantor sets and of all Lebesgue measure zero sets are homeomorphic to the Hilbert space $\ell_2$. Moreover, we investigate the hyperspace $CL_H(R)$ of all nonempty closed subsets of the real line $R$ with the Hausdorff (infinite-valued) metric. We show that a nonseparable component of $CL_H(R)$ is homeomorphic to the Hilbert space $\ell_2(2^{\aleph_0})$ as long as it does not contain any of the sets $R, [0,\infty), (-\infty,0]$.