# Self-homeomorphism group

From Topospaces

*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.