assignment3

# assignment3 - X(3 If y ∈ D x,r then D x,r = D y,r(4 If D...

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

MATH 255 ASSIGNMENT 3 This assignment is due in class on Monday, January 31 Problems Please justify carefully your answers. 1. [10 points] Is it true that in any metric space ( X,d ), cl( D ( x,r )) = { y X : d ( x,y ) r } . Provide a proof or ﬁnd a counterexample. 2. [10 points] Let ( X,d ) be a metric space and S 1 ,S 2 X . Prove that cl( S 1 S 2 ) = cl( S 1 ) cl( S 2 ) . 3. [15 points] Let ( X,d ) be a metric space and S X . (1) Prove that ( ∂S ) ∂S . (2) Find an example such that the inclusion in (1) is proper. (3) Suppose that S is a closed set. Prove that ( ∂S ) = ∂S . 4. [15 points] Let X be the collection of all sequences of positive integers. If x = ( n j ) j =1 and y = ( m j ) j =1 are two elements of X , set k ( x,y ) = inf { j : n j 6 = m j } and d ( x,y ) = ( 0 if x = y 1 k ( x,y ) if x 6 = y. In the assignment 2 you have shown that d is a metric on X . Prove the following: (1) d ( x,z ) max { d ( x,y ) ,d ( y,z ) } . (2) Any open ball D ( x,r ) is a closed subset 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.

Unformatted text preview: X . (3) If y ∈ D ( x,r ), then D ( x,r ) = D ( y,r ). (4) If D ( x,r 1 ) ∩ D ( y,r 2 ) 6 = ∅ , then either D ( x,r 1 ) ⊂ D ( y,r 2 ) or D ( y,r 2 ) ⊂ D ( x,r 1 ). 5. [20 points] Let [ a,b ] be an interval and X = C ([ a,b ]) the vector space of all continuous real-valued functions on [ a,b ] equipped with the norm k f k = max a ≤ t ≤ b | f ( t ) | . Let F be a ﬁxed continuous function such that F ( t ) > 0 for all t ∈ [ a,b ] and let S = { f ∈ X : | f ( t ) | < F ( t ) for all t ∈ [ a,b ] } . 1 (1) Prove that S is open. (2) Find cl( S ). (3) Find ∂S . Remark. If you have diﬃculties solving this problem you should attend the tutorial (the problem will be solved there). 2...
View Full Document

## This document was uploaded on 02/22/2011.

### Page1 / 2

assignment3 - X(3 If y ∈ D x,r then D x,r = D y,r(4 If D...

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

View Full Document
Ask a homework question - tutors are online