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.

showing it is not possible for $(0,1)$ is easy, just take the open cover $(0,1-\frac{1}{n})$.
Showing it is possible for $[0,1]$ is not so easy, you may want to look here for proofs that do not go through the whole Heine-Borel theorem.
