Unformatted text preview: R such that K 1 ⊃ K 2 ⊃ K 3 ⊃ ... . Prove that there exists at least one point x ∈ R such that x ∈ K n for all n ∈ N ; that is, the intersection T ∞ n =1 K n is not empty. Suppose that T K n is empty. Since K n is compact, it is closed. Thus, K c n is open. Since T K n is empty, we have that S K c n = R . In particular, { K c n } ∞ n =2 is an open cover for K 1 . Thus, there are a ﬁnite number of indices α 1 ,...,α N so that K 1 ⊆ K c α 1 ∪ ··· ∪ K c α N = ( K α 1 ∩ ··· ∩ K α N ) c . This means that K 1 ∩ K α 1 ∩ ··· ∩ K α N is empty. But by the nexting property, if α 1 ≤ ··· ≤ α N , then K 1 ∩ K α 1 ∩ ··· ∩ K α N = K α N which was nonempty by assumption. 1...
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 Spring '10
 MetCalfe
 Calculus, Topology, Empty set, kN, General topology, finite subcollection, nonempty compact sets

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