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HW18solns

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

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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 6 = 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 6 = z , then either x 6 = y or y 6 = 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 S U α where the U α are open. Then p is in one of the U α , say U α 0 . Since U α 0 is open, there is an > 0 such that B ( p ; ) U α 0 S U α . Hence S U α is open. 4. Consider S = { p 1 , . . . , p n } in some metric space X . We will show that S C is
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