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!