I know that it's a theorem by Moise that every compact 3-manifold admits a finite triangulation but to me the astounding part of that statement is the existence part instead of the finite one. So I was wondering: Is there an easier (possibly more modern) way to show that every compact, triagulable 3-manifold already admits a finite triangulation? All attempts I've made so far end up with me choosing a metric but I'd rather avoid that.
Suppose you have a compact space $X$ and a triangulation (or more generally, a CW complex structure) of $X$. Choose one point in the interior of each simplex, and let $S\subseteq X$ be the set of all these points. Then $S$ is closed, because its intersection with each closed simplex is finite and hence closed (it contains only one point in the interior of each face of the simplex). Thus $S$ must be compact. But the exact same argument shows that any subset $A\subset S$ is also closed, so $S$ has the discrete topology. Thus $S$ must be finite, so the triangulation of $X$ can have only finitely many simplices.