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.,
G × X → X , ( g , x ) ↦ g ⋅ x is a continuous map. Together with the group action, X is called a G-space.
If f : H → G is a continuous group homomorphism of topological groups and if X is a G-space, then H can act on X by restriction: h ⋅ x = f ( h ) x , 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 G → 1 (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 X H for the set of all x in X such that h x = x . For example, if we write F ( X , Y ) for the set of continuous maps from a G-space X to another G-space Y, then, with the action ( g ⋅ f ) ( x ) = g f ( g − 1 x ) , F ( X , Y ) G consists of f such that f ( g x ) = g f ( x ) ; i.e., f is an equivariant map. We write F G ( X , Y ) = F ( X , Y ) G . Note, for example, for a G-space X and a closed subgroup H, F G ( G / H , X ) = X H .
