Mathematical Thinking: Problem-Solving and Proofs (2nd Edition)

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Unformatted text preview: 1. We simply verify the properties of a metric. Positive definite: This is just a trivial observation from the definition. Symmetric: Obviously d(x, x) = d(x, x). If x = y, then d(x, y) = 1 = d(y, x). Triangle Inequality: If x = z, then d(x, z) = 0 = 0+0 d(x, y)+d(y, z). If x = z, then either x = y or y = z. In either case we have d(x, z) = 1 = 1 + 0 d(x, y) + d(y, z). 2. Let x (a, b). Set = min {x - a, b - x}. Then B(x; ) = (x - , x + ) (x - (x - a), x + (b - x)) = (a, b). Therefore (a, b) is open. 3. Let p U where the U are open. Then p is in one of the U , say U0 . Since U0 is open, there is an > 0 such that B(p; ) U0 U . Hence U is open. 4. Consider S = {p1 , . . . , pn } in some metric space X. We will show that S C is open. Let x S C and set = min {d(x, p1 ), . . . , d(x, pn )}. We claim B(x; ) S C . Let y B(x; ). By the Triangle Inequality, d(y, pi ) d(x, pi ) - d(x, y) > - = 0, so y = pi and thus y S C . It follows that S C is open, which means S is closed. 5. (a) C x U x U x U for every C U C x U for every x (b) C x U x U x U for some C U C x U for some x 1 ...
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This document was uploaded on 03/25/2011.

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