Yahoo Answers: Answers and Comments for Prove that if F is closed subset of a compact set K in R then F is compact bu using the definition below...... [Mathematics]
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Wed, 12 Mar 2008 01:35:57 +0000
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Yahoo Answers: Answers and Comments for Prove that if F is closed subset of a compact set K in R then F is compact bu using the definition below...... [Mathematics]
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From Pascal: Let A be an arbitrary open cover of F. Since F...
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https://answers.yahoo.com/question/index?qid=20080312013557AAgLFj9
Wed, 12 Mar 2008 02:40:15 +0000
Let A be an arbitrary open cover of F. Since F is closed, R\F is open, so A∪{R\F} is an open cover of all of R, thus of K. Since K is compact, A contains a finite subcover of K, say C. Then for every element x∈F, we have x∈K (since F⊆K), so ∃S∈C such that x∈S (as C covers K). Moreover, since x∈F, x∉R\F, so S≠R\F, thus S∈C\{R\F}. It follows that C\{R\F} covers F. However, C\{R\F} ⊆ (A∪{R\F})\{R\F} = A, so C\{R\F} is a finite subcover of A. Since we can find such a subcover for any open cover, F is compact. Q.E.D.