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S27 - 4th January 2005 Munkres 27 Ex 27.1(Morten Poulsen...

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4th January 2005 Munkres § 27 Ex. 27.1 (Morten Poulsen). Let A X be bounded from above by b X . For any a A is [ a, b ] compact. The set C = A [ a, b ] is closed in [ a, b ], hence compact, c.f. theorem 26.2. The inclusion map j : C X is continuous, c.f. theorem 18.2(b). By the extreme value theorem C has a largest element c C . Clearly c is an upper bound for A . If c A then clearly c is the least upper bound. Suppose c / A . If d < c then ( d, ) is an open set containing c , i.e. A ( d, ) = , since c is a limit point for A , since c C A . Thus d is not an upper bound for A , hence c is the least upper bound. Ex. 27.3. (a) . K is an infinite, discrete, closed subspace of R K , so K can not be contained in any compact subspace of R K [Thm 28.1]. (b) . The subspaces ( -∞ , 0) and (0 , + ) inherit their standard topologies, so they are connected. Then also their closures, ( -∞ , 0] and [0 , + ) and their union, R K , are also connected [Thm 23.4, Thm 23.3]. (c) . Since the topology R K is finer than the standard topology [Lemma 13.4] on R we have U is connected in R K Ex 23.1 U
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