solutionshw3-555-2011

solutionshw3-555-2011 - Solutions homework 3 Page 69...

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Solutions homework 3. Page 69 Problem 10: a. Assume A and B are neigborhoods of x . Then there exist r 1 > 0 such that U r 1 ( x ) A and r 2 > 0 such that U r 2 ( x ) B . let r = min { r 1 ,r 2 } . Then r > 0 and U r A B . Hence A B is a neighborhood of x . b. Let B r ( c ) be a closed ball and x n B r ( c ) such that x n x . Then from last weeks homework we get d ( c,x n ) d ( c,x ). Now d ( c,x n ) r for all n implies that also d ( c,x r . Hence x B r ( c )). Thus B r ( c ) is closed. That every open ball is open was proved in class. c. Analog of 4.3.3: If A subset of a metric space X , then x / ¯ A ⇐⇒ x ( A c ) . Proof: x / ¯ A ⇐⇒ there exists r > 0 such that U r ( x ) A = ∅ ⇐⇒ there exists r > 0 such that U r ( x ) A c ⇐⇒ x ( A c ) . Analog of 4.3.5 If A subset of a metric space, then i) A is open ⇐⇒ A c is closed and ii) A is closed ⇐⇒ A c is open. Proof: ii) follows immediately from the analog
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This note was uploaded on 02/05/2012 for the course MATH 555 taught by Professor Staff during the Spring '07 term at South Carolina.

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solutionshw3-555-2011 - Solutions homework 3 Page 69...

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