104_Fa08_hw_5_sol - Homework 5 for MATH 104 Due: Thursday...

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Unformatted text preview: Homework 5 for MATH 104 Due: Thursday Oct 9 Problem 1 Pugh, p.115, #5 Solution. ( ) Let U M be open. Let x U . Then there exists an r-neighborhood of x that is completely contained in U . This means that no sequence in M \ U can get closer than r to x , which implies that x is not a limit point of M \ U . ( ) Assume U M is not open. Then there exists x U such that for every r > 0, M r ( x ) contains a point from M \ U . By letting r = 1 / n for n = 1 , 2 , 3 , . . . , we construct a sequence ( x n ) such that d ( x , x n ) < 1 / n and x n M \ U . Then x n x , so x is a limit point of M \ U . Problem 2 If ( M , d ) is a metric space, define the distance ( A , B ) of two non-empty sets A , B M as ( A , B ) = inf { d ( x , y ) : x A , y B } . Give an example of two closed subsets of R (with respect to the standard metric) such that ( A , B ) < d ( x , y ) for all x A , y B . Explain your answer....
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104_Fa08_hw_5_sol - Homework 5 for MATH 104 Due: Thursday...

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