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204Lecture62008Web

204Lecture62008Web - Economics 204 Lecture 6Monday August 4...

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Economics 204 Lecture 6–Monday, August 4, 2008 Section 2.8, Compactness Definition 1 A collection of sets U = { U λ : λ Λ } in a metric space ( X, d ) is an open cover of A if U λ is open for all λ Λ and λ Λ U λ A (Λ may be finite, countably infinite, or uncountable.) A set A in a metric space is compact if every open cover of A contains a finite subcover of A . In other words, if { U λ : λ Λ } is an open cover of A , there exist n N and λ 1 , · · · , λ n Λ such that A U λ 1 ∪ · · · ∪ U λ n It is important to understand what this definition does not say. In particular, it does not say “ A has a finite open cover;” note that every set is contained in X , and X is open, so every set has a cover consisting of exactly one open set. Like the ε - δ definition of continuity, in which you are given an arbitrary ε > 0 and are challenged to specify an appropriate δ , here you are given an arbitrary open cover and challenged to specify a finite subcover of the given open cover. Example: (0 , 1] is not compact in E 1 . To see this, let U = U m = 1 m , 2 : m N Then m N U m = (0 , 2) (0 , 1] 1

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Given any finite subset { U m 1 , . . . , U m n } of U , let m = max { m 1 , . . . , m n } Then n i =1 U m i = U m = 1 m , 2 (0 , 1] so (0 , 1] is not compact. Note that this argument does not work for [0 , 1]. Given an open cover { U λ : λ Λ } , there must be some λ Λ such that 0 U λ , and therefore U λ [0 , ε ) for some ε > 0, and a finite number of the U m ’s we used to cover (0 , 1] would cover the interval ( ε, 1]. This is not a proof that [0 , 1] is compact, since we need to show that every open cover has a finite subcover, but it is suggestive, and we will soon see that [0 , 1] is indeed compact. Example: [0 , ) is closed but not compact. To see that [0 , ) is not compact, let U = { U m = ( 1 , m ) : m N } Given any finite subset { U m 1 , . . . , U m n } of U , let m = max { m 1 , . . . , m n } Then U m 1 ∪ · · · ∪ U m n = ( 1 , m ) [0 , ) Theorem 2 (8.14)

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