In topology, a continuous group action on a topological space X is a group action of a topological group G that is continuous: i.e., is a continuous map.
Together with the group action, X is called a G-space.
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 as 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 continuous maps from a G-space X to another G-space Y, then, with the action
Note, for example, for a G-space X and a closed subgroup H,
This topology-related article is a stub.