Self-homeomorphism group

This article defines an association of a group to every topological space. The association is not necessarily functorial

Definition

The self-homeomorphism group of a topological space is defined in the following equivalent ways:

• It is the group of homeomorphisms from the topological space to itself, under composition
• It is the automorphism group of the topologicla space, viewed as an object in the category of topological space with continuous maps

The association of a self-homeomorphism group to a topological space is not functorial.

Related notions

• Mapping class group is the quotient of the self-homeomorphism group, by the group of self-homeomorphisms isotopic to the identity map.