I have some confusion on the definition of discontinuous action
I found the definition on nlab
It is written that
The action of a topological group $G$ on a topological space $X $ is called properly discontinious if every point $x \in X$ has a nbd $U_x$ such that the intersection $g(U_x) \cap U_x$ with its translate under the group action via some element $g \in G$ is nonempty only for the neutral element $e \in G$
$g(U_x) \cap U_x \neq \emptyset \implies g=e$
My confusion : What is the meaning of $g(U_x) ?$
My thinking : I think $g(U_x) $ is a function.I think it is a identity fucntion $g(U_x)=U_x$
But here im not getting why $g(U_x)=U_x$ become identity function ?