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

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.