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 de±ned on X × X . a) Prove that the union of arbitrarily many open sets is an open set. b) Prove that the intersection of ±nitely 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 in±nitely 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 de±ned on X × X . Say whether each of the following statements is true or not. If the statements
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This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at University of California, Berkeley.

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

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