Prove this action is properly discontinuous..

Question

Consider the group $\mathbb Z_2=\{0, 1\}$ acting on the sphere $\mathbb S^n$ through the group actions $\psi_0=Id$ and $\psi_1=-Id$. Show this actions is properly discontinuos? The definition of properly discontinuity I'm using is: Let $G$ a group and $\psi_g$ an action on a topological space $X$, then $\psi_g$ is said to be properly discontinuous if for every $x\in X$ there is an open neighbourhood $U$ of $x$ such that $U\cap \psi_g(U)=\phi$ for all $g\in G$ with $g\neq e$.

Answer 1

Hints:

(1) Clearly the group is discrete...

(2) The inverse image of any compact subset of $\,S^n\,$ is compact...which is also clear.

The above already makes it prop. disc., but with your definition perhaps the way is a little longer:

Since the only non-trivial element of the group maps points on the sphere to its antipodes (=opposite points), you can take even a huge open neighborhood of any $\,x\in S^n\,$ , say half the half sphere determined by the midpoint of the sphere's diameter joining $\,x\,,\,-x\,$, and then clearly $ 1\cdot U\cap U=\emptyset\,$ , as $\,1\cdot U\,$ will be all the opposite (antipode) points of $\,U\,$ ...