Compactness of the open and closed unit intervals In the article by Tao it's explained that the compactness can be formulated in the most general way as:
(All open covers have finite subcovers) If $`V_\alpha:\alpha\in\mathcal{a}`$ is any collection of open sets which covers $X$, then there must exist a finite sub-collection $V_{\alpha_1},V_{\alpha_2}...V_{\alpha_k}$ of these sets which still cover $X$.
Question: How is it possible to show (both intuitively and rigorously) that such coverage by a finite collection of open sets is possible for $X=[0,1]$ and not possible for $X=(0,1)$?
Edit: a related question and another one on open covers.

I think the most rigorous way of showing this more generally is using the Heine-Borel theorem, which says that a subset of $\mathbb{R}^n$ is compact if and only if it is closed and bounded. Maybe this is less intuitive but if you understand the proof of the theorem it gives you a large insight. 
