I am currently re-reading a course on basic algebraic topology, and I am focussing
on the parts that I feel I had very little understanding of. There is one exercise
in the chapter devoted to groups acting on topological spaces (preceding the chapter
on covering spaces) that I have spent several frustrating hours on, but I just can't
crack.
I will state the question below, but PLEASE, DO NOT POST A SOLUTION OR A HINT. I am
only interested in knowing wether the statement in question is true. I have looked
on the internet for a reference about this fact, but didn't get anywhere. If you
know a book that gives proves it, or a document online where this is discussed, I
would like to get the reference, in case I continue to fail at giving a proof of
this the next week.
I recall some terminology first: let $X,Y$ be two topological spaces, and
$f:X\rightarrow Y$ a map that isn't supposed to be continuous. The author defines
such a map to be $\mathrm{PROPER}$ whenever the following two properties are
satisfied : $f$ is a closed map, and $\forall y\in Y, ~f^{-1}(\lbrace y\rbrace)$ is
a compact subspace of $X$. It then follows that for all compact subsets $K$ of
$Y,~f^{-1}(K)$ is a compact subset of $X$. Also, if $X$ is Hausdorff, and $Y$ is a
locally compact Hausdorff, then properness is equivalent to this property.
Let $X$ be a topological space, and $G$ a topological group. Suppose there is a
continuous left group action $\rho: G\times X\rightarrow X,~(g,x)\mapsto g\cdot x$.
Let $\theta:G\times X\rightarrow X\times X, ~(g,x)\mapsto (x,g\cdot x)$. The author
defines the group action to be $\mathrm{proper}$ if $\theta$ is a proper map.
Here is the question: "Show the action of a compact Hausdorff group $G$ on a
Hausdorff space $X$ is always proper".
As I said, I have struggled with this for days (since friday). IS THAT STATEMENT
TRUE? There are no further hypothesis, $X$ is not supposed to be locally compact,
and the action is completely arbitrary (continuous of course, but not supposed free,
or other things).
Thank you for your time!
Best Answer
Since that's all you want to hear in terms of mathematics: Yes, it's true.
You can find the argument in Bourbaki's Topologie générale, Chapitre III. More precisely, it's the statement of Proposition 2 a) of §4 in Ch. III on page TG III.28 of my edition.
Some time ago, I wrote a résumé on proper actions on MO, which you can find here.