Path-Connectedness of the Hyperspace of Compact Subsets of $\mathbb{R}^n$

TL;DR

This paper provides an elementary proof that the hyperspace of all compact subsets of b^n, equipped with the Hausdorff metric, is path-connected, making the concept accessible to undergraduates.

Contribution

It offers a simple, accessible proof of the path-connectedness of the hyperspace of compact subsets of b^n using basic topological and vector space ideas.

Findings

The hyperspace b(b^n) is path-connected.

Elementary proof relies on vector structure and basic metric topology.

Accessible to students with undergraduate-level topology knowledge.

Abstract

When one considers the collection $\mathcal{H}(\mathbb{R}^n)$ of all compact subsets of $\mathbb{R}^n$ and equip it with a topology, many questions can be asked about the topological space one ends up with. This is an example of a hyperspace, a mathematical object which has been studied in a more abstract setting since the beginning of the 20th century. Here we give an elementary proof of the path-connectedness of $\mathcal{H}(\mathbb{R}^n)$, with the topology induced by the Hausdoff metric, by exploring the vector structure of $\mathbb{R}^n$ and using only basic ideas of topology of metric spaces that undergraduate students with just a basic knowledge of these concepts will be able to understand.