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Macroeconomics Exam Review 7

Macroeconomics Exam Review 7 - Solutions for Foundations of...

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1.85 1. Let { 𝐺 𝑖 } be a (possibly infinite) collection of open sets. Let 𝐺 = 𝑖 𝐺 𝑖 . Let 𝑥 be a point in 𝐺 . Then there exists some particular 𝐺 𝑗 which contains 𝑥 . Since 𝐺 𝑗 is open, 𝐺 𝑗 is a neighborhood of 𝑥 . Since 𝐺 𝑗 𝐺 , 𝑥 is an interior point of 𝐺 . Since 𝑥 is an arbitrary point in 𝐺 , we have shown that every 𝑥 𝐺 is an interior point. Hence, 𝐺 is open. What happens if every 𝐺 𝑖 is empty? In this case, 𝐺 = and is open (Exercise 1.81). The other possibility is that the collection { 𝐺 𝑖 } is empty. Again 𝐺 = which is open. Suppose { 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑛 } is a finite collection of open sets. Let 𝐺 = 𝑖 𝐺 𝑖 . If 𝐺 = , then it is trivially open. Otherwise, let 𝑥 be a point in 𝐺 . Then 𝑥 𝐺 𝑖 for all 𝑖 = 1 , 2 , . . ., 𝑛 . Since the sets 𝐺 𝑖 are open, for every 𝑖 , there exists an open ball 𝐵 ( 𝑥, 𝑟 𝑖 ) 𝐺 𝑖 about 𝑥 . Let 𝑟 be the smallest radius of these open balls, that
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