This is Exercise II.3 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to Approach0, it is new to MSE.
The Details:
From Mac Lane and Moerdijk, p. 37
Definition 1: Given an object $C$ in the category $\mathbf{C}$, a sieve on $C$ [. . .] is a set $S$ of arrows with codomain $C$ such that
$f \in S$ and the composite $fh$ if defined implies $fh \in S$.
From p. 70 ibid. . . .
Definition 2: A sieve $S$ on $U$ is said to be a covering sieve for $U$ when $U$ is the union of all the open sets $V$ in $S$.
From the exercise ibid. . . .
Definition 3: An action $G\times X\stackrel{\cdot}{\to}X$ of a group $G$ on a space $X$ is said to be proper if for every point $x\in X$ there exists a neighbourhood $U_x$ of $x$ with the property that $g\cdot U_x\cap U_x\neq \varnothing$ implies $g = 1$ for all $g\in G$.
From p. 82 ibid. . . .
Definition 4: A covering map $p: \stackrel{\sim}{X} \to X$ is a continuous map between topological spaces such that each $x\in X$ has an open neighborhood $U$, with $x\in U \subset X$, for which $p^{-1}U$ is a disjoint union of open sets $U_i$, each of which is mapped homeomorphically onto $U$ by $p$.
The Question:
Prove that if $G$ acts properly on $X$, the quotient map $\pi: X\to X/G$ to the space $X/G$ of orbits is a covering.
Thoughts:
Let $G$ act properly on a topology $X$, denoted $(g, x)\mapsto g\cdot x$.
My first thought is to consider finite $G$ or finite $X$ to make the problem tractable. To this end, let $G$ be the trivial group.
Then, indeed, for all $x\in X$, we have that, since $X\in\mathcal{O}(X)$, there exists a neighbourhood (namely, $X$), for which, for any $g\in G$, if $g\cdot X\cap X\neq \varnothing$, then $g=1$. This is far from edifying though. One can see clearly that $X/G$ is in bijection with $X$ (I think) and, indeed, all open sets are subsets of $X$. Thus $\pi:X\to X/G$, as described in the exercise, is a covering.
But I'm not sure about that last sentence. Does it even make sense?
Does one have to consider $\pi$ and $\{\pi\}$ as roughly the same thing to fit into Definition 1 and Definition 2?
I feel way off.
Where do I go from here?
Please help 🙂
Best Answer
Hint: Suppose that the action $G\times X\to X$ is proper in the sense of this book (which is a nonstandard definition by the way). Given $x\in X$, show that $V=\pi(U_x)$ is a neighborhood of $y=\pi(x)$ in $Y=X/G$. Now, describe the preimage $\pi^{-1}(V)$ as a disjoint union of suitable open subsets of $X$. Analyze the restriction of $\pi$ to these open subsets. ... Conclude that $\pi: X\to Y$ is a covering map.