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.,
is a continuous map. Together with the group action, "X" is called a "G"-space.
If formula_2 is a continuous group homomorphism of topological groups and if "X" is a "G"-space, then "H" can act on "X" "by restriction": formula_3, 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 formula_4 (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 formula_5 for the set of all "x" in "X" such that formula_6. For example, if we write formula_7 for the set of continuous maps from a "G"-space "X" to another "G"-space "Y", then, with the action formula_8,
formula_9 consists of "f" such that formula_10; i.e., "f" is an equivariant map. We write formula_11. Note, for example, for a "G"-space "X" and a closed subgroup "H", formula_12.