# hw3 - Y,d Y ) are both compact metric spaces. Prove that X...

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Modern Analysis, Homework 3 Due June 22, 2010 Directions: hand in all of problems 0-5, but please try problem 6. 0. [12] (2 pts) Rudin, Ch 2, problem 12. 1. [12] (2 pts) Rudin, Ch 2, problem 14. 2. [10] (6 pts) Rudin, Ch 2, problem 9. 3. [25] (5 pts) Rudin, Ch 2, problem 23. 4. [25] (5 pts) Rudin, Ch 2, problem 24. 5. [30] (5 pts) Rudin, Ch 2, problem 26. 6. [30] (optional, not necessary!) Let ( X,d X ) and ( Y,d Y ) be metric spaces. Deﬁne the product metric d X × Y to be the map d X × Y : ( X × Y ) × ( X × Y ) R given by d X × Y (( x,y ) , ( x 0 ,y 0 )) = p d X ( x,x 0 ) 2 + d Y ( y,y 0 ) 2 . a) (1 pt) Prove that d X × Y is a metric on X × Y . b) (5 pts) Suppose that ( X,d X ) and (
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Unformatted text preview: Y,d Y ) are both compact metric spaces. Prove that X × Y is compact, when given the product metric. Hint: You can prove this directly by looking at open subsets of X × Y , but it may be easier to the result of problem 5: a metric space is compact if and only if and only if every inﬁnite subset of it has a limit point. If you try it this way, start building a limit point one component at a time. If you work directly from the deﬁnition by open covers, look at “slices” in one coordinate. 1...
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## This note was uploaded on 10/18/2011 for the course MATH S4061X taught by Professor Peters during the Spring '11 term at Columbia.

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