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On page 44-45 of the notes from http://www.msc.uky.edu/droyster/courses/fall99/math4181/classnotes/notes5.pdf, the author writes in the proof that the interval $[0,1]$ is compact that supposing the interval isn't compact, that this implies either $[0,1/2]$ or $[1/2,1]$ isn't compact. How do we know that both $[0,1/2]$ and $[1/2,1]$ are not compact?

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Perhaps my questions was not clear enough, I understand that supposing [0,1] is not compact implies some interval inside [0,1] is not compact. What I am confused about is why can't [0,1/2] and [1/2,1] both not be compact. – Chris Oct 5 '12 at 18:48
It says either or, that means one or the other or both. – Graphth Oct 5 '12 at 18:51
Thank you, my eyes seem to have glossed over that. – Chris Oct 5 '12 at 18:52
I put italic using '_ qzerty_' instead of '$$', and changed tags. – Davide Giraudo Oct 5 '12 at 20:23

2 Answers

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Suppose $[0,\frac{1}{2}]$ and $[\frac{1}{2},1]$ are compacts. Now take any open cover of $[0,1]$. This open cover is an open cover of $[0,\frac{1}{2}]$ and $[\frac{1}{2},1]$. Now you use the compactness of $[0,\frac{1}{2}]$ and $[\frac{1}{2},1]$ and conclude that $[0,1]$ is compact.

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The union of two compact sets is compact: An open cover of $A\cup B$ contains a finite subcover of $A$ and a finite subcover of $B$, together a finite subcover of $A\cup B$.

(And Hausdorff is clear, too).

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