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royden9 - ROYDEN REAL ANALYSIS 3RD ED CHAPTER 9 c c Problem...

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ROYDEN, REAL ANALYSIS 3RD ED. CHAPTER 9 Problem 9-2. { K c n : n 1 } and 0 form an open covering of K 1 . Suppose N 1 K c n i O K 1 . Then K c n N O K 1 K n N . Hence O K n N . Problem 9-4. Let A, B be closed subsets. Note that A, B are compact. Fix x A . By Problem 9.3 there are disjoint open sets O x and U x with x O x and B U x . Since A is compact, we can find { O x i : 1 i N } which covers A . Let O A = S n 1 O x i and O B = T n 1 U x i . Problem 9-16. Urysohn’s Lemma and Probem 9.4 imply g is one-to-one. Now use Proposition 9.5 and Tychonoff Theorem. Problem 9-19 a. By Problem 9.18 there is an open set O K with ¯ O compact. By Problem 9.4 and Urysohn’s Lemma there is a continuous function f on ¯ O with f | K = 1 and f | ¯ O \ O = 0. Extend f by setting f = 0 on ¯ O c . Hence f | O c = 0. Now use Proposition 8.3 with A = O c and B = ¯ O to show f is continuous on X . Note that { x : f ( x ) 6 = 0 } ⊆ O ¯ O is thus compact. b. Suppose K = T 1 O i . By Problem 9.18 we may assume each ¯ O i is compact. Otherwise consider T 1 ( O i O ), where O is given in Problem 9.18. By Part a there are continuous functions f i with 0 f i 1 , f i | K = 1 and f i | O c i = 0 . Let g = 1 f i / 2 i . Then g is continuous as the series converges uniformly. If x / K then x O c i for some i . Hence f i ( x ) = 0 and then g ( x ) < 1. Problem 9-26. We need the following result (p.238, Dugundji): Typeset by A M S -T E X 1
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2 Lemma. Assume X is locally compact. Then for any x and open set O 3 x , there exists an open set V
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