I'm reading James' Fibrewise topology book and I'm trying to understand the proof of proposition 7.4 , it says:
Let X be a proper G-space . Then X is fibrewise regular over X/G.
Proof For any $x \in X$ and $U$ a neighbourhood of x:
It begins saying the stabilizer $G_x$ of x is compact since $ \theta^{-1}(x,x) = G_x \times \{ x \} $ and so there exists a neighbourhood N of $G_x$ and $V'$ of x in X such that $N \times V' \subset \theta^{-1}(U \times_{X/G} U)$ ...
- Compactness is obvious since X is a proper G-space means by definition $\theta : G\times X \rightarrow X \times_{X/G} X $ given by $\theta(g,x) = (x,gx)$ is proper. However, why do those neighbourhoods N and V' exist?
Without loss of generality, assume that $U$ is open. Then $W := \theta^{-1}(U \times_{X/G} U)$ is open, and contains $G_x\times \{x\}$. Then, for every $g\in G_x$, there are open neighbourhoods $N_g$ of $g$ and $V_g$ of $x$ such that $N_g \times V_g \subset W$. Take a finite subcover of $\{N_g : g\in G_x\}$, and let
$$N = \bigcup_{k=1}^m N_{g_k}; \qquad V' = \bigcap_{k=1}^m V_{g_k}.$$
By construction, $N\times V' \subset W$.