- #1
BWV
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- 1,769
Is this statement accurate? (Not trying a proof, just trying to understand the concept)
[0,1] is compact because any infinite collection of open subsets that covers the set has to include the boundry points therefore there is a finite collection of these sets that also covers
(0,1) is not compact because to cover the set you need an infinite number of subsets to obtain the limit condition on the open boundary - with a finite collection of open subsets you can always arbitrarily find a point outside of the union of your finite collection of open subsets and closer to a boundary
[0,1] is compact because any infinite collection of open subsets that covers the set has to include the boundry points therefore there is a finite collection of these sets that also covers
(0,1) is not compact because to cover the set you need an infinite number of subsets to obtain the limit condition on the open boundary - with a finite collection of open subsets you can always arbitrarily find a point outside of the union of your finite collection of open subsets and closer to a boundary