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. 

It's not possible.  If a finite set $F = \{x_1,\ldots,x_n\}$ is covered by a family of sets $\mathcal{S}$ (open or otherwise) then it is covered by a finite subfamily.  Just pick $S_1 \in \mathcal{S}$ containing $x_1$, $S_2 \in \mathcal{S}$ containing $x_2$ and so forth.
