# 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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online