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solutions-assignment3

# solutions-assignment3 - MATH 255 ASSIGNMENT 3 short...

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MATH 255 ASSIGNMENT 3, short solutions Problems Please justify carefully your answers. 1. [10 points] Is it true that in any metric space ( X, d ), cl( D ( x, r )) = { y X : d ( x, y ) r } . Provide a proof or find a counterexample. Solution. Counterexample. Let X be a set that contains more then 2 points and let d ( x, y ) = 0 if x = y , d ( x, y ) = 1 if x 6 = y . We have shown in class that any subset of X is both open and closed. Clearly, D ( x, 1) = cl( D ( x, 1)) = { x } . On the other hand, { y X : d ( x, y ) 1 } = X . If X has more then 2 points, X 6 = { x } . 2. [10 points] Let ( X, d ) be a metric space and S 1 , S 2 X . Prove that cl( S 1 S 2 ) = cl( S 1 ) cl( S 2 ) . Solution. It is an immediate consequence of the definition of the closure that A B cl( A ) cl( B ) (you should write a short justification). Hence, cl( S 1 ) cl( S 1 S 2 ), cl( S 2 ) cl( S 1 S 2 ), and so cl( S 1 ) cl( S 2 ) cl( S 1 S 2 ) . To prove the opposite inclusion, note that S 1 cl( S 1 ), S 2 cl( S 2 ), imply S 1 S 2 cl( S 1 ) cl( S 2 ). Since cl( S 1 ) cl( S 2 ) is a closed set and cl( S 1 S 2

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solutions-assignment3 - MATH 255 ASSIGNMENT 3 short...

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