Using the "open cover" definition of compactness to show that $[0,1]$ is compact

Asked Oct 17 '15 at 04:59

Active Oct 17 '15 at 08:19

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How can I prove that the interval $[0,1]$ is compact in real line topology?

I know how to prove it using concept of boundedness and closedness, but I wish to understand it by using the "open cover" definition of compactness: a set is compact if its every open cover has a finite subcover.

Kindly help me to see the compactness topologically.

edited Oct 17 '15 at 08:19

Taumatawhakatangihangakoauauot

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asked Oct 17 '15 at 04:59

Nitin Uniyal

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did you open up a topology book (e.g., Munkres) and read the proof? – Ittay Weiss Oct 17 '15 at 05:01
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This is called the Heine-Borel theorem, and there's a proof in every single topology textbook, and I'm guessing countless more on the web. Search away. – stochasticboy321 Oct 17 '15 at 05:05
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All of the proofs there are worth looking at, but the one that you probably want is this one. – Brian M. Scott Oct 17 '15 at 18:04

Using the "open cover" definition of compactness to show that $[0,1]$ is compact

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