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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.
If f : H → G {\displaystyle f:H\to 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 x {\displaystyle h\cdot x=fx} , 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 G → 1 {\displaystyle G\to 1}
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 {\displaystyle X^{H}} for the set of all x in X such that h x = x {\displaystyle hx=x}. For example, if we write F {\displaystyle F} for the set of continuous maps from a G-space X to another G-space Y, then, with the action = g f {\displaystyle =gf} , F G {\displaystyle F^{G}} consists of f such that f = g f {\displaystyle f=gf} ; i.e., f is an equivariant map. We write F G = F G {\displaystyle F_{G}=F^{G}}. Note, for example, for a G-space X and a closed subgroup H, F G = X H {\displaystyle F_{G}=X^{H}}.