Wikipedia defines strongly continuous group action as follows:
A group action of a topological group G on a topological space X is said to be
strongly continuous if for all x in X, the map g ↦ g.x is continuous with
respect to the respective topologies.
My issue with this is that with this definition, every continuous group action is strongly continuous:
Suppose $G$ acts continuously on the space $X$. Then for all $x\in X$, the map $g\mapsto g\cdot x$ can be decomposed as
$$G \xrightarrow{\;r_x\;} G\times X \xrightarrow{\;\alpha\;} X, $$
where $\alpha$ is multiplication and $r_x\colon g\mapsto (g,x)$. But $\alpha$ is continuous by definition, and it's clear that $r_x$ is continuous (if $U\times V$ is a cylinder set in $G\times X$, then $r_x^{-1}(U\times V)=U$, which is open). Thus, the map $g\mapsto g \cdot x$ is continuous for all $x\in X$. Hence, by the definition above, the action of $G$ on $X$ is strongly continuous.
Am I missing something?
Best Answer
That's a feature, not a bug. Strong continuity is actually supposed to be a weaker condition than continuity (see also strong operator topology, which is weaker than the norm topology). For example, the action of $\mathbb{R}$ on $L^2(\mathbb{R})$ by translation is strongly continuous if $L^2(\mathbb{R})$ is given the norm topology but not continuous.