Let $G$ be a topological group acting continuously on a topological space $X$. This means that $G \times X \rightarrow X$ is a continuous function.
A continuous map $Y \rightarrow Z$ is said to be proper if the preimage of a compact set is compact.
If $G$ acts continuously on $X$, then the action is said to be proper if the map
$$G \times X \rightarrow X \times X, (g,x) \mapsto (x, g\cdot x)$$
is proper.
I would have expected the definition of a proper group action to be that $G \times X \rightarrow X$ is proper, not $G \times X \rightarrow X \times X$. Why is the definition the way it is? What is the most natural way to think about this?
There are two situations in which I have encountered this notion which I want to understand better:
1 . An analytic Lie group $H$ over a local field of characteristic zero is acting properly and freely on an analytic manifold $X$, and for each $x \in X$, the map $h \mapsto h.x$ is an immersion $H \rightarrow X$. Then some result in Bourbaki, Differential and Analytic Manifolds says that the quotient space $H \setminus X$ has the natural structure of an analytic manifold.
2 . There is an arrangement $\mathscr H$ of hyperplanes in a real affine space $E$, and $W$, the group of affine transformations in $E$ generated by the reflections about the hyperplanes, is assumed to stabilize the set of hyperplanes $\mathscr H$ and act properly on $E$. This is the situation for many results described in Chapter V of Bourbaki, Lie Groups and Lie Algebras.
Best Answer
Observe that for a point $x_0\in X$, the pre-image of the compact subset $\{x_0\}\subset X$ under the action map $G\times X\to X$ is $\{(g,g^{-1}\cdot x_0):g\in G\}\subset G\times X$. If the action map is proper, then this is a compact subset of $G\times X$. Since the restriction of the first projection to this subset defines a continuous surjection onto $G$, the group $G$ itself has to be compact. Thus requiring the action map to be proper brings you back to the case of compact groups.