Math 301 Problem Set 8 Solutions

Math 301 Problem Set 8 Solutions - Math 301/ENAS 513: HW 8...

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Math 301/ENAS 513: HW 8 Solution Triet Le Remark 1 . Let U be any collection of subsets of a generic set S . Then ± [ U ∈U U ! c = ² U ∈U U c . 1: If K = { K α } is a collection of compact subsets of a metric space ( S,d ) such that the intersection of every finite subcollection of K is nonempty, then T K α ∈K K α is nonempty. In particular, if { K n } is a sequence of nonempty closed and bounded sets in R n such that K n K n +1 , for n 1. Then n =1 K n is not empty. Solution 1 . Suppose that T K α ∈K K α = , then there exists a compact set K ∈ K such that for any x K , x / T K α 6 = K K α . In other words, K T ³ T K α 6 = K K α ´ = . This implies K S K α 6 = K K α . Hence U = { K c α : K α 6 = K } is an open cover for K . Since K is compact, there exists a finite open subcover, { K α i : i = 1 ,...,n } , such that K n [ i =1 K c α i , K α i ∈ K . By the remark, we kave = K ² ± n [ i =1 K c α i ! c = K ² ± n ² i =1 K c α i ! . This contradicts the assumption that every finite collection of K is nonempty.
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Math 301 Problem Set 8 Solutions - Math 301/ENAS 513: HW 8...

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