Macroeconomics Exam Review 7

Macroeconomics Exam Review 7 - 1.85 1. Let { } be a...

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Unformatted text preview: 1.85 1. Let { } be a (possibly infinite) collection of open sets. Let = . Let be a point in . Then there exists some particular which contains . Since is open, is a neighborhood of . Since , is an interior point of . Since is an arbitrary point in , we have shown that every is an interior point. Hence, is open. What happens if every is empty? In this case, = and is open (Exercise 1.81). The other possibility is that the collection { } is empty. Again = which is open. Suppose { 1 , 2 ,..., } is a finite collection of open sets. Let = . If = , then it is trivially open. Otherwise, let be a point in . Then for all = 1 , 2 ,..., . Since the sets are open, for every , there exists an open ball ( ,...
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This note was uploaded on 01/16/2012 for the course ECO 2024 taught by Professor Dr.dumond during the Fall '10 term at FSU.

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