204Lecture62009web

204Lecture62009web - Economics 204 Lecture 6Monday, August...

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Economics 204 Lecture 6–Monday, August 3, 2009 Section 2.8, Compactness Defnition 1 A collection of sets U = { U λ : λ Λ } in a metric space ( X, d )isan open cover of A if U λ is open for all λ Λand λ Λ U λ A (Λ may be Fnite, countably inFnite, or uncountable.) Ase t A in a metric space is compact if every open cover of A contains a Fnite subcover of A .I no t h e r words, if { U λ : λ Λ } is an open cover of A ,thereex ist n N and λ 1 , ··· n Λ such that A U λ 1 ∪···∪ U λ n It is important to understand what this defnition does not say. In particular, it does not say “ A has a fnite 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 defnition 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 fnite subcover oF the given open cover. Example: (0 , 1] is not compact in E 1 . To see this, let U = ± U m = ² 1 m , 2 ³ : m N ´ Then m N U m =(0 , 2) (0 , 1] 1
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Given any fnite 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 ² 6⊇ (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 )Forsome ε> 0, and a fnite 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 fnite 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 fnite subset { U m 1 m n } oF U m { m 1 n } Then U m 1 ∪···∪ U m n 1 ) 6⊇ [0 , ) Theorem 2 (8.14) Every closed subset A of a compact metric space ( X, d ) is compact.
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204Lecture62009web - Economics 204 Lecture 6Monday, August...

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