Continuous group action
# 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 $(g\cdot f)(x)=gf(g^{-1}x)$, $F(X,Y)^{G}$ consists of f such that $f(gx)=gf(x)$; 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, $F_{G}(G/H,X)=X^{H}$.

