Let's choose an open covering for $\left [ 0,1 \right ]$. For example $$\left \{ \left ( \frac 1 n,1-\frac 1 n \right ) \mid n\in \{ 3,4,\dots\} \right \}.$$ How can one choose a finite open subcover to prove compactness?
Showing that $[0,1]$ is compact
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general-topology
compactness
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4If your collection of open sets is to be a cover of the interval, it has to cover 0 and 1, which - even given the fact that my browser is not interpreting your formula - seem to be missing from the union of your proposed covering sets. – 2012-08-30
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5Indeed, that is an open cover of $(0,1)$, not $[0,1]$. Since the former interval is not compact, it is not a surprise that there is no finite subcover. – 2012-08-30
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0See here for some other proofs of this fact: http://math.stackexchange.com/questions/368108/how-to-prove-every-closed-interval-in-r-is-compact – 2016-04-23
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0Read the statement of compactness theorem again. WORD BY WORD. – 2018-12-19