Editing Group action (section)

=== Actions of topological groups ===
{{Main|Continuous group action}}
Now assume {{math|''G''}} is a [[topological group]] and {{math|''X''}} a topological space on which it acts by homeomorphisms. The action is said to be ''continuous'' if the map {{math|''G'' × ''X'' → ''X''}} is continuous for the [[product topology]].

The action is said to be ''{{visible anchor|proper}}'' if the map {{math|''G'' × ''X'' → ''X'' × ''X''}} defined by {{math|(''g'', ''x'') ↦ (''x'', ''g''&sdot;''x'')}} is [[proper map|proper]].{{sfn|tom Dieck|1987|loc=}} This means that given compact sets {{math|''K'', ''K''′}} the set of {{math|''g'' ∈ ''G''}} such that {{math|''g''&sdot;''K'' ∩ ''K''′ ≠ ∅}} is compact. In particular, this is equivalent to proper discontinuity if {{math|''G''}} is a [[discrete group]].

It is said to be ''locally free'' if there exists a neighbourhood {{math|''U''}} of {{math|''e''<sub>''G''</sub>}} such that {{math|''g''&sdot;''x'' ≠ ''x''}} for all {{math|''x'' ∈ ''X''}} and {{math|''g'' ∈ ''U'' &setminus; {{mset|''e''<sub>''G''</sub>}}}}.

The action is said to be ''strongly continuous'' if the orbital map {{math|''g'' ↦ ''g''&sdot;''x''}} is continuous for every {{math|''x'' ∈ ''X''}}. Contrary to what the name suggests, this is a weaker property than continuity of the action.{{citation needed|date=May 2023}}

If {{math|''G''}} is a [[Lie group]] and {{math|''X''}} a [[differentiable manifold]], then the subspace of ''smooth points'' for the action is the set of points {{math|''x'' ∈ ''X''}} such that the map {{math|''g'' ↦ ''g''&sdot;''x''}} is [[smooth map|smooth]]. There is a well-developed theory of [[Lie group action]]s, i.e. action which are smooth on the whole space.