Yahoo Answers: Answers and Comments for If F is aclosed subset of acompact set K in R then F is compact? [Mathematics]
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From shosho N
enUS
Mon, 10 Mar 2008 10:05:43 +0000
3
Yahoo Answers: Answers and Comments for If F is aclosed subset of acompact set K in R then F is compact? [Mathematics]
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From holdm: Let {Oi} be an open cover of F. Then {Oi + F...
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Mon, 10 Mar 2008 10:16:43 +0000
Let {Oi} be an open cover of F. Then {Oi + F} is an open over of K, since F closed implies F open. Therefore there exists a finite subcover S of {Oi+ F}. Remove F and this is a finite subcover of F. For, let x be in F. Then x is in some open set in S. But x is not in F. Therefore x is in some Oi in S. and the Oi's of S form a finite subcover of F.
Therefore F is compact.

From squeek: Since K is compact in R, it is closed and boun...
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Mon, 10 Mar 2008 10:12:04 +0000
Since K is compact in R, it is closed and bounded. Therefore, F must also be bounded.
Since F is a closed and bounded subset of R, then F is compact.