I would like to understand the following statement:
The usual way to study the action of a group is to construct a continuous section of the
action.
Take a group $G$ acting on a topological set $S$. A continuous section of the action $\mathcal{A} : G\times S \rightarrow S$ is a section of the projection $p:S \rightarrow S/G$ (continuous for the topology on $S$).
$S/G$ is the set all of orbits of $S$ under the action of $G$ which form a partition of $X$.
So if I construct a section $s:S/G \rightarrow S$ such that $p\circ s=id_{S/G}$, how to relate to the study of the action $\mathcal{A}$.