Yahoo Answers: Answers and Comments for How to show K is compact? [Mathematics]
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enUS
Sat, 03 Jul 2010 22:09:16 +0000
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Yahoo Answers: Answers and Comments for How to show K is compact? [Mathematics]
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From Steiner: Suppose K is not compact. Then, by Heine Borel...
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Sun, 04 Jul 2010 05:48:04 +0000
Suppose K is not compact. Then, by Heine Borel theorem, K is not bounded or not closed.
If K is not bounded, define f(x) = x, x ∈ K. Then, f is continuous. And since K is unbounded, for every M > 0 there is x ∈ K such that f(x) = x > M, which shows f is a continuous unbounded function from K to R.
If K is not closed, then K has a limit point a that doesn't belong to K. So, the function from K to R given by f(x) = 1/(x  a is well defined and continuous. Since a is a limit point of K, then for every M > 0 we can find x ∈ K such that x  a < 1/M, which implies that f(x) > M. Hence, f is unbounded.
This shows that, if K is not compact, then there is a continuous f:K → R that's not bounded. By contraposition, the claim is proved.

From villagran: considering ok and L are closed they're co...
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Wed, 07 Dec 2016 09:25:56 +0000
considering ok and L are closed they're compact (considering X is compact). permit x be a factor no longer in ok. considering X is metric this is Hausdorff and as a result there exists 2 disjoint open contraptions Ux and Vx such that ok is a subset of Ux and x is contained in Vx. The set of all contraptions Vx, the place x is in L covers L. as a result there exists a finite subcover V1,...,Vn. permit U1,...,Un be the corresponding open contraptions from until eventually now. permit U be the intersection of U1,...,Un and F the union of V1,...,Vn. this is elementary to work out that U and V are as needed.