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psAnalysis2Qu - Fall 2007 Problem Set#02 Second Analysis...

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Fall 2007 ARE211 Problem Set #02 Second Analysis Problem Set Due date: Sep 18 Problem 1 For the following problem, consider an arbitrary universe X, together with an arbitrary metric d defined on X × X . a) Prove that the union of arbitrarily many open sets is an open set. b) Prove that the intersection of finitely many open sets is an open set. c) Identify what part of your proof in part (b) would no longer hold if you would consider infinitely many sets. d) Prove that the intersection of arbitrarily many closed sets is closed. (This is another very short proof, you just have to come up with a little trick. Hint: use DeMorgans formula i I ( X \ A i ) = X \ ( i I A i ) ) (Note: The union of two sets A, B is: A B = { x X | x A x B } ). The intersection of two sets A, B is: A B = { x X | x A x B } ). Problem 2 For the following problem, consider an arbitrary universe X, together with an arbitrary metric d defined on X × X . Say whether each of the following statements is true or not. If the statements are true, prove them. If the statements are wrong, give a counter-example.
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