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Unformatted text preview: 2.103 Assume is uhc and be any closed set in . By Exercise 2.97 ( ) = [ + ( ) ] is open. By the previous exercise, + ( ) is open which implies that ( ) is closed. Conversely, assume ( ) is closed for every closed set . Let be an open subset of so that is closed. Again by Exercise 2.97, + ( ) = [ ( ) ] By assumption ( ) is closed and therefore + ( ) is open. By the previous exercise, is uhc. The lhc case is analogous. 2.104 Assume that is uhc at . We first show that ( ) is bounded and hence has a convergent subsequence. Since ( ) is compact, there exists a bounded open set containing ( ). Since is uhc, there exists a neighborhood of such that ( ) for . Since , there exists some such that...
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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.
- Fall '10