245Chw1sol

245Chw1sol - Math 245C Spring 2007 Homework 1 Solutions: I...

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Unformatted text preview: Math 245C Spring 2007 Homework 1 Solutions: I read problems 10, 26, 43, 49, and 74. Here are correct solutions to those and to problem 13, which is needed for Problem 49. 10. (a.) If A X is open and closed (called clopen by some) and negationslash = A negationslash = X then A and X \ A are nonempty disjoint open sets with union X , so that X is not connected. If X is disconnected, then X = U V with U V = , and U,V both proper. Then A = U is clopen and negationslash = A negationslash = X . (b.) Assume X = uniontext A E = U V where U V = , negationslash = U,V negationslash = X, and U and V are (relatively) open in X . Then for each , U E and V E are disjoint relatively open subsets of E with ( U E ) ( V E ) = E . Since E is connected it follows that E = U E or E = V E , and since U negationslash = and V negationslash = , each alternative holds for at least one A. Take U A such that E U U, and V A such that E V V. Then intersectiontext A E E U E V = , a contradiction. (c.) Suppose not. Then there are open sets U 1 and U 2 such that A = ( U 1 U 2 ), A U i negationslash = for i = 1 , 2, and A U 1 U 2 = . By the definition of closure, A U i negationslash = for i = 1 , 2, and by the above A ( U 1 U 2 ), and A U 1...
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245Chw1sol - Math 245C Spring 2007 Homework 1 Solutions: I...

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