# hw6 - Math508, Fall 2010 Jerry L. Kazdan Problem Set 6 D UE...

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Math508, Fall 2010 Jerry L. Kazdan Problem Set 6 DUE: Thurs. Oct. 28, 2010. Late papers will be accepted until 1:00 PM Friday . 1. Give examples of the following: a) An open cover of { x R : 0 < x 1 } that has no ﬁnite sub-cover. b) A metric space having a bounded inﬁnite sequence with no convergent subsequence. c) A metric space that is not complete. 2. Let K be a compact set in a metric space M and let p M be a point not in K . Deﬁne the distance dist ( p , K ) between p and K as dist ( p , K ) = inf x K d ( p , x ) . a) Show there is at least one point q K that has this minimum distance, so d ( p , q ) = dist ( p , K ) b) Is there a unique such point q ? Proof or counterexample. c) Is the assertion in part a) still true if you only assume that K is a closed (but not compact) subset of R 2 ? Proof or counterexample. 3. For any two sets S , T in a metric space, deﬁne the distance between these sets as dist ( S , T ) = inf x S , y T d ( x , y ) . Assume both

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## This note was uploaded on 03/06/2012 for the course MATH 508 taught by Professor Staff during the Fall '10 term at UPenn.

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hw6 - Math508, Fall 2010 Jerry L. Kazdan Problem Set 6 D UE...

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