104_Fall09_review_mt_2_sol - Partial Answers to Review...

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Unformatted text preview: Partial Answers to Review Problems for 2nd Midterm 1. (a) If M is a metric space and S M , then S \ S is open. TRUE: If S \ S is empty, S \ S is open. Otherwise, let x S \ S . S = S S c is the set of all x M such that every r-neighborhood of x contains points from S and S c . Since x 6 S , there must exist an r-neighborhood M r ( x ) of x that is fully contained in S . M r ( x ) cannot contain a point from S , since then it would also contain a point from the complement of S . (b) [0 , 1] \ { 1 n : n N } is a compact subset of R . FALSE: By the Heine-Borel Theorem, a subset of R is compact iff it is closed and bounded. But the given set is not closed since e.g. 1 is a limit point of the set that is not in the set. (c) There exists a continuous mapping from [0 , 1] onto Q . FALSE: [0 , 1] is connected and the images of connected sets under continuous mappings are connected. But Q is disconnected....
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104_Fall09_review_mt_2_sol - Partial Answers to Review...

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