204Lecture62008Web

# 204Lecture62008Web - Economics 204 Lecture 6Monday, August...

This preview shows pages 1–4. Sign up to view the full content.

Economics 204 Lecture 6–Monday, August 4, 2008 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 08/01/2008 for the course ECON 204 taught by Professor Anderson during the Fall '08 term at University of California, Berkeley.

### Page1 / 11

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online