Final Solution

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Unformatted text preview: x < 52, let = (x - 2)2. If x 52 then let = (3 - x)2. For x (2, 52) we need to show that B(x-2)2 (x) = ((x + 2)2, (3x - 2)2) (2, 3). This follows since (x + 2)2 > 2 x > 2. Similarly, (3x - 2)2 < 3 x < 83. By the same token, we need to show that for x [52, 3) that B(3-x)2)(x) = ((3x - 3)2, (x + 3)2) (2, 3). Naturally, (3x - 3)2 > 2 x > 73 and (x + 3)2 < 3 x < 3 which is true. (c) Since X is not closed, X is not compact. (b) X c = (-, 2] [3, ). Note that for x = 2 X c , no > 0 so that B (2) X c . Hence, X c is not open and so X is not closed. 5. Consider the set X = [a, b] R where - < a < b < . (a) Is X closed? Why or why not? (b) Is X open? Why or why not? (c) Is X compact? Why or why not? 6 (d) Find the closure of X. Solution: (a) Since X c = (-, a) (b, ) is the union of two open sets, hence X c is open. Since X c is open, X is closed. (c) The set X is compact since i...
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This document was uploaded on 03/20/2014 for the course ECON 2p30 at Brock University, Canada.

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