MATH 301 Homework 7 - ( x n ) has a cluster point in K x ....

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Math 301, Homework 7 1. For each positive a 2 R ; Let I a = a 3 ; 3 a ± : (a) Let e U be the collection e U = f I a : 0 < a < 3 g . Find all integers k so that e U is an open cover for A k = ( k;k + 3) : (b) For which of the sets A k above, does e U (c) Which of the sets A k above are compact? 2. For a set M R 2 ; x -axis to be the set in R M x = f x : there is some y so that ( x;y ) 2 M g : (a) Find a set M; so that M x is open (in R ) but M is not open (in R 2 ) : (b) Suppose K is a compact set in R 2 . Prove that K x is also compact in R : . A is open in R , then A ± R is open in R 2 " ; then start with an open cover C of K x and create a cover for K : Or: star with a sequence ( x n ) in K x and use that K is sequentially compact to show that
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Unformatted text preview: ( x n ) has a cluster point in K x . (c) Find a closed set N in R 2 such that N x is not closed (in R ) . 3. Suppose K 1 ;K 2 ;:::;K m are compact sets. prove that K = K 1 [ K 2 [ ::: [ K m is compact in two ways: (a) Start with a cover for K and &nd a &nite subcover. (b) Start with a sequence in K and &nd a cluster point. 4. Problems on Continuity: (a) Let f : [0 ; 1] ! R be de&ned as f ( x ) = ² x if x 2 Q if x 62 Q : Use the & ² ± de&nition to prove that f is continuous at x = 0 : Is it continuous anywhere else? (b) Let f : [0 ; 1] ! R be de&ned as f ( x ) = ² 1 if x 2 Q if x 62 Q : Prove that f is not continuous anywhere. 5. 1...
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This note was uploaded on 06/25/2010 for the course MATH MATH 301 taught by Professor Alberterkip during the Fall '08 term at Sabancı University.

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