21504hw2sol - MATH 215 Homework 2 Turn in by until 9:40 a.m Each question is 4 points total 20 points 1 For the following sets E nd E and int(E Are

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June 17, 2004 MATH 215 Homework 2 Turn in by June 22, 2004 until 9:40 a.m. Each question is 4 points, total: 20 points. 1. For the following sets E find E 0 and int( E ). Are these sets open, closed, neither? a) In ( R , | · | ), E = { ( - 1) n + (1 /m ) : n,m N } b) In ( R 2 ,d 2 ), E = { ( x,y ) : 1 x 2 + y 2 < 4 } . Solution. a) Since ( - 1) n is either +1 or - 1, the set E = {- 1 + 1 1 , - 1 + 1 2 , - 1 + 1 3 , - 1 + 1 4 ,... } ∪ { 1 + 1 1 , 1 + 1 2 , 1 + 1 3 , 1 + 1 4 ,... } . Then E 0 = {- 1 , 1 } , int( E ) = . E is neither open nor closed. b) E is an annulus (i.e. the region between two concentric circles where the inner boundary is included, but the outer boundary is excluded). E 0 = { ( x,y ) : 1 x 2 + y 2 4 } , int( E ) = { ( x,y ) : 1 < x 2 + y 2 < 4 } . E is neither open nor closed. 2. Prove that in a metric space the union of an arbitrary collection of open sets is open. Proof. Let { G α : α A } be an arbitrary (finite, countable or uncountable) collection of open subsets of
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This note was uploaded on 04/15/2010 for the course INDUSTRIAL ie513 taught by Professor Zeynephuygur during the Spring '10 term at Bilkent University.

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21504hw2sol - MATH 215 Homework 2 Turn in by until 9:40 a.m Each question is 4 points total 20 points 1 For the following sets E nd E and int(E Are

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