Continuous group action

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.,
 * $$G \times X \to X, \quad (g, x) \mapsto g \cdot x$$

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

If $$f: H \to G$$ is a continuous group homomorphism of topological groups and if X is a G-space, then H can act on X by restriction: $$h \cdot x = f(h) x$$, 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 $$G \to 1$$ (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 $$X^H$$ for the set of all x in X such that $$hx = x$$. 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 $$(g \cdot f)(x) = g f(g^{-1} x)$$, $$F(X, Y)^G$$ 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, $$F_G(G/H, X) = X^H$$.