assignment3

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

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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
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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 xed 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 diculties solving this problem you should attend the tutorial (the problem will be solved there). 2...
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assignment3 - X . (3) If y D ( x,r ), then D ( x,r ) = D (...

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