In example 3.4.2, the author shows that if $G$ is compact, then every $G$-space is proper.
I have a question: According to the definition, we need to show that every compact subset $K$ of $X\times X$, the inverse image of $\varphi$ is compact. But every compact subset of $X\times X$ is not of the form $C\times C$, where $C$ is compact in $X$. Why can the proof consider only compact subsets of this form?
First of all, there appears a background assumption that all the spaces mentioned are locally compact Hausdorff; without this assumption many of the statements in the page shown are false.
Now given this assumption (actually, it suffices to just know $X$ is Hausdorff), suppose $K\subseteq X\times X$ is compact and let $p,q:X\times X\to X$ be the two projections. Then $p(K)$ and $q(K)$ are compact and thus $C=p(K)\cup q(K)$ is also compact. Observe now that $K\subseteq C\times C$, and $K$ is closed in $C\times C$ since it is compact and $X$ is Hausdorff. Thus $\varphi^{-1}(K)$ is closed in $\varphi^{-1}(C\times C)$. So, if we know $\varphi^{-1}(C\times C)$ is compact, then $\varphi^{-1}(K)$ is also compact.