# Continuous group action

In topology, a **continuous group action** on a topological space *X* is a group action of a group *G* that is continuous: i.e.,

is a continuous map. Together with the group action, *X* is called a ** G-space**.

If is a continuous group homomorphism of topological groups and if *X* is a *G*-space, then *H* can act on *X* *by restriction*: , making *X* a *H*-space. Often *f* is either an inclusion or a quotient map. In particular, any topological space may be thought of a *G*-space via (and *G* would act trivially.)

Two basic operations are that of taking the space of points fixed by a subgroup *H* and that of forming a quotient by *H*. We write for the set of all *x* in *X* such that . For example, if we write for the set of continuous maps from a *G*-space *X* to another *G*-space *Y*, then, with the action ,
consists of *f* such that ; i.e., *f* is an equivariant map. We write . Note, for example, for a *G*-space *X* and a closed subgroup *H*, .

## References

- John Greenlees, Peter May,
*Equivariant stable homotopy theory*

## See also