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?
Understanding Proof that Interval $[0,1]$ is Compact.
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real-analysis
general-topology
compactness
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0I put italic using '_ qzerty_' instead o$f$ '$$', and changed tags. – 2012-10-05
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).