# Spaces of Homeomorphisms

Chapter

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## 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.

## Copyright information

© The Author(s) 2018