Spaces of Homeomorphisms

Abstract

In this chapter, we study some topological properties of the space H(X), the set of all homeomorphisms from a metric space X onto itself, where H(X) has either the uniform topology or the fine topology. In particular, we study the countability and connectedness of the space H(X) with the uniform and fine topologies. Also for the case that $X=\mathbb {R}^n$, three different natural compatible metrics are used to generate three different uniform topologies on $H(\mathbb {R}^n)$. These three homeomorphism spaces are shown to be not homeomorphic to each other for $n>1$, and are also compared to $H(\mathbb {R}^n)$ with the fine, point-open and compact-open topologies.