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Unformatted text preview: ) B , then A open = A B open . Let a A and b B . If b 6 = 0, and ( a, a + ) A for some > 0, then ( ab  b  , ab +  b  ) A B. Now suppose b = 0 and (, ) B for some > 0. If a A satises a 6 = 0, then ( a  ,  a  ) A B , providing a neighborhood of ab = 0 contained in A B in that case. If a = 0, then there exists > 0 with (, ) A , and then (, ) A B . 4.1.5.2 The set A is closed because A = A 1 A 2 A n , where A i is the set dened by f i ( x ) = 0, and each A i is the inverse image of the closed set { } by a continuous map, hence closed. It need not be compact, of course: consider the case n = 1 and f ( x ) = sin x , so A = Z . 1...
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This homework help was uploaded on 02/23/2008 for the course MATH 4130 taught by Professor Hubbard during the Fall '07 term at Cornell University (Engineering School).
 Fall '07
 HUBBARD
 Sets

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