Let $G$ be a topological group act continuously on a topological space $X$. Why the continuity of the action of $G$ on $X$ implies that $G$ embedded as topological group in $S_{X}$. Here $S_{X}$ is the symetric group on $X$ i.e the group of all self bijection on $X$ equipped with the pointwise topology.
Why the map $G\longrightarrow S_{X}$ is injective if the action of G on X is continuous? Also, why this map is an homeomorphism?
Thank for any help.
That's the way an action $\alpha : G \times X \to X$ is defined.
For a topological group, the group operation must be continuous. We define $\phi : G \to S_X$, where $\phi : g \mapsto \alpha_g$, by $\alpha_g(x) := \alpha(g,x)$ for all $g \in G$ and $x \in X$. Since $\alpha$ is a group action, we have $\alpha(g,\alpha(h,x)) = \alpha(gh,x)$ for all $g,h \in G$ and $x \in X$. Hence $\alpha_g \circ \alpha_h \equiv \alpha_{gh}$.
This tells us that $\phi$ is a group homomorphism.
