TITLE: Finite sub cover for $(0,1)$
QUESTION [13 upvotes]: While learning topology one learns about compact set. The standard definition is:

A set $X$ is said to be compact if open cover has a finite subcover.

Since $[0,1]$ is compact, if we take a open cover for this we should be able to get a finite subcover. I know, that $(0,1)$ is not compact, so there must exists some open cover for $(0,1)$ which doesn't admit any finite subcover. But how does one prove this fact?

REPLY [6 votes]: Alternatively, use the fact that every compact subspace of a Hausdorff space is closed. Since (0,1) is not closed, it cannot be compact. (this is theorem 26.3 in Munkres)