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 ${\displaystyle f:H\to G}$ is a continuous group homomorphism of topological groups and if X is a G-space, then H can act on X by restriction: ${\displaystyle h\cdot x=f(h)x}$, making X a H-space. Often f is either an inclusion or a quotient map. In particular, any topological space may be thought of a G-space via ${\displaystyle G\to 1}$ (and G would act trivially.)

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