mth536_problem_8

mth536_problem_8 - MTH536 Chapter 8 Gene Quinn – p. 1/...

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Unformatted text preview: MTH536 Chapter 8 Gene Quinn – p. 1/ Chapter 8 If ¤ is a collection of topologies for a set § ¨¦ ¢ £¡ , then is a topology for . ¥ ¢ ¢ £¡ ¥ – p. 2/ Proof of Proposition For criteria: §¦ to be a topology for     ¥  ¦ , it must satisfy the following three ¢¡ ©  ¦ ¥        ©   $¦ !   " #  % " #     "& ! ¦ ¦ – p. 3/ Proof of Proposition i Since every is a topology, by definition and therefore to ¢ '¡ § ¨¦ ¢ ¢ '¡ and  belong to each ¥ ¢¡ – p. 4/ Proof of Proposition ii Suppose  ( (  $ 0 1)  ¢ £¡ Then since each is a topology for   ©  , $ ) ¢¡ which implies that     © ¢ ¢ '¡ § ¦ ¢ '¡ ¥ – p. 5/ Proof of Proposition iii Suppose " # ¤ is a an arbitrary collection of open sets in Then " #  $ % ¢ '¡  §¦ . which by the definition of a topology implies that " 2  $ ¢ 3¡ ) and therefore " #  " ¢ ¢ '¡ § ¦ " ) ¢¡ © – p. 6/ ...
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This note was uploaded on 03/23/2010 for the course MATH 515 taught by Professor Staff during the Spring '08 term at Iowa State.

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