# Self-homeomorphism group

From Groupprops

This article defines a natural context where a group occurs, or is associated, with another algebraic, topological or analytic structure

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## Definition

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

- The elements of this group are homeomorphisms from the topological space to itself and the multiplication is by composition
- It is the automorphism group of the topological space when viewed as an object in the category of topological spaces with continuous maps

The self-homeomorphism group of any topological space can be given the structure of a topological group using the compact-open topology.

## Facts

For a homogeneous space, the topological space can be viewed, set-theoretically, as a quotient of the self-homeomorphism group by the isotropy at any point (because there is only one orbit). This follows from the fundamental theorem of group actions. (Note: Please don't confuse iso*tropy* (the stabilizer) with iso*topy*).

Particular examples are manifolds, which are always homogeneous.