G × X → X , ( g , x ) ↦ g ⋅ x {\displaystyle G\times X\to X,\quad (g,x)\mapsto g\cdot x}
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 Xby restriction: h ⋅ x = f ( h ) x {\displaystyle h\cdot 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 as a _G_-space via G → 1 {\displaystyle G\to 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 {\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 ( X , Y ) {\displaystyle 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 ) {\displaystyle (g\cdot f)(x)=gf(g^{-1}x)} , F ( X , Y ) G {\displaystyle F(X,Y)^{G}} consists of f such that f ( g x ) = g f ( x ) {\displaystyle f(gx)=gf(x)} ; i.e., f is an equivariant map. We write F G ( X , Y ) = F ( X , Y ) G {\displaystyle 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 {\displaystyle F_{G}(G/H,X)=X^{H}} .
