Question about finite sets/compactness I understand that every finite subset of a metric space is compact. But are there any topological spaces where finite sets are not compact? Is that even possible? I don't think it is but I just want to be sure. 

If you know the general definition of compactness, this isn't too hard to prove. Let's do so thoroughly:
Proposition Let $X$ be a finite topological space, $S=(s_1,...,s_n)$ a finite subset. Then $S$ is compact.
Proof $\ $ Let $\mathcal{U}=\{U_\alpha\}$ be an open cover of $S$. Then each $U_\alpha$ is open and each $s_i\in U_\alpha$ for some $\alpha$, say $U_{\alpha_i}$. Then $(U_{\alpha_1},...,U_{\alpha_n})$ is a finite subcover of $S$.
