hw8sol - Math 125A, Fall 2009 Solutions for Homework 8...

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Unformatted text preview: Math 125A, Fall 2009 Solutions for Homework 8 (Prepared by Matt Low) 1. (a) Suppose S = B i where each B i is an open ball. For each x in S , there exists a ball B i such that x B i . This ball B i is in the form of B r i ( x i ) of radius r i > 0 and center x i X . Let r = r i- d ( x,x i ) > 0. Then the open ball B r ( x ) is contained in B i , and hence is also contained in S . That is, each point x S is an interior point of S and thus S is open. (b) Let S be an open set in ( X,d ). This means that, for every x S , there exists an open ball B r x ( x ), of radius r x centered at x , such that B r x ( x ) S . Now, we claim that: S = [ x S B r x ( x ) Indeed, if y S , then y B r y ( y ), and B r y ( y ) appears as a term in the union S x S B r x ( x ), so that y S x S B r x ( x ). On the other hand, if y S x S B r x ( x ), then y B r x ( x ) for some x S . Because B r x ( x ) S , it then follows that y S . Thus S =...
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This note was uploaded on 03/18/2012 for the course MATH 67 taught by Professor Schilling during the Fall '07 term at UC Davis.

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hw8sol - Math 125A, Fall 2009 Solutions for Homework 8...

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