1

I want to compare three definitions for an action $G\times X\to X$ to be proper:

  1. For each $x\in X$, there is a neighborhood $U\subseteq X$ such that $gU\cap U=\emptyset$ for all but finitely many $g\in G$.
  2. For each $x,x'\in X$, there are neighborhoods $U,U'\subseteq X$ such that $gU\cap U'=\emptyset$ for all but finitely many $g\in G$.
  3. For each compact subset $K\subseteq X$, we have $gK\cap K=\emptyset$ for all but finitely many $g\in G$.

The equivalence of (2) and (3) holds for locally compact $X$, see e.g. Lee's Introduction into topological manifolds. However, I don't see if the first definition is equivalent to the other ones and under which conditions ($X$ hausdorff?). Obviously, (2) implies (1). I think I could show the converse if I knew that $$\{(x,gx);\,x\in X\text{ and }g\in G\}\subseteq X\times X$$ is closed if (1) holds. Is this clear?

FKranhold
  • 779

1 Answers1

4

First of all, you seem to be thinking of $G$ as a discrete group. For example, if $G$ is not discrete, then the last phrase in (3) would have to be replaced by "all $g$ outside of a compact subset of $G$."

Assuming $G$ is discrete and $X$ is a locally compact Hausdorff space, (2) and (3) are both equivalent to the action being proper, as you noted. But (1) is definitely weaker. Consider the following example: Let $X=\mathbb R^2\smallsetminus\{(0,0)\}$, and define an action of $\mathbb Z$ on $X$ by $n\cdot (x,y) = (2^n x, 2^{-n} y)$. This satisfies your condition (1), but it's not a proper action. The subset $K \times K \subseteq X\times X$ is compact, where $K = \{(x,y): \max(|x|,|y|)=1\}$, but $\rho^{-1}(K\times K)$ contains the sequence $(n, (2^{-n},1))$, which has no convergent subsequence.

For more about these various definitions, see this answer and this one.

Jack Lee
  • 50,850