mth536_problem_8_3

mth536_problem_8_3 - MTH536 Chapter 8 Problem 3 Gene Quinn...

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Unformatted text preview: MTH536 Chapter 8 Problem 3 Gene Quinn – p. 1/ Chapter 8 Problem 4 Prove that a set set such that is open if and only if, given £ ¢ ¡ ¤ £ ¡ ¢ , there is an open – p. 2/ ¥ Proof of Proposition Suppose every is contained in some open set ¦ § ©  §£ ¤ § ¨£ ¤ ¡ ¢ . Then – p. 3/ ¥ Proof of Proposition Suppose every is contained in some open set ¦ By hypothesis, every the ’s, so § belongs to ©  §£ ¤ and therefore to the union of §£ § ©  §£ ¤ §£ ¡ ¢ § ¨£ ¤ ¡ ¢ . Then – p. 3/ ¥ Proof of Proposition Suppose every is contained in some open set ¦ By hypothesis, every the ’s, so § belongs to ©  §£ ¤ and therefore to the union of §£ Together with the previous result, this establishes that § © Since open, is a union of open sets, and arbitrary unions of open sets are is open. § ©  §£   §£ ¤ §£ ¡ ¢ § ¨£ ¤ ¡ ¢ . Then – p. 3/ ¥ Proof of Proposition Now suppose is open. For any ¦ ¡ ¤ £ Then is open and , and therefore £ ¡ ¢ £ §£  ¡ ¢ ¢ , choose – p. 4/ ...
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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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mth536_problem_8_3 - MTH536 Chapter 8 Problem 3 Gene Quinn...

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