Heine-Borel theorem. I'm interested in one question about Heine-Borel theorem.
We know that if S is bounded and closed then it's compact.
Standard proof using fact that some segment is compact. And because of S is bounded it's have left and right bound of segment. But my question is my S should be closed? We could take open , bounded S and some segment which cover out set. Why it's so necessary ?

Let $S=(0,1)$ and consider the cover $\{(1/n,1) : n \ge 2\}$. The union of any finite subset of the cover is of the form $(1/n,1)$ for some $n$, so no finite subset can cover $S$.
