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Macroeconomics Exam Review 9

Macroeconomics Exam Review 9 - c 2001 Michael Carter All...

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1 2 1 𝑆 𝐵 1 / 2 ((2 , 0)) Figure 1.2: Open ball about (2 , 0) relative to 𝑋 1.95 Assume 𝑆 is connected. Suppose 𝑆 is not an interval. This implies that there exists numbers 𝑥, 𝑦, 𝑧 such that 𝑥 < 𝑦 < 𝑧 and 𝑥, 𝑧 𝑆 while 𝑦 / 𝑆 . Then 𝑆 = ( 𝑆 ( −∞ , 𝑦 )) ( 𝑆 ( 𝑦, )) represents 𝑆 as the union of two disjoint open sets (relative to 𝑆 ), contradicting the assumption that 𝑆 is connected. Conversely, assume that 𝑆 is an interval. Suppose that 𝑆 is not connected. That is, 𝑆 = 𝐴 𝐵 where 𝐴 and 𝐵 are nonempty disjoint closed sets. Choose 𝑥 𝐴 and 𝑧 𝐵 . Since 𝐴 and 𝐵 are disjoint, 𝑥 = 𝑧 . Without loss of generality, we may assume 𝑥 < 𝑧 . Since 𝑆 is an interval, [ 𝑥, 𝑧 ] 𝑆 = 𝐴 𝐵 . Let 𝑦 = sup { [ 𝑥, 𝑧 ] 𝑆 } Clearly 𝑥 𝑦 𝑧 so that 𝑦 𝑆 . Now 𝑦 belongs to either 𝐴 or 𝐵 . Since 𝐴 is closed in 𝑆 , [ 𝑥, 𝑧 ] 𝐴 is closed and 𝑦 = sup { [ 𝑥, 𝑧 ] 𝑆
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