This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Homework 5 for MATH 104 Due: Thursday Oct 9 Problem 1 Pugh, p.115, #5 Solution. ( ) Let U M be open. Let x U . Then there exists an rneighborhood of x that is completely contained in U . This means that no sequence in M \ U can get closer than r to x , which implies that x is not a limit point of M \ U . ( ) Assume U M is not open. Then there exists x U such that for every r > 0, M r ( x ) contains a point from M \ U . By letting r = 1 / n for n = 1 , 2 , 3 , . . . , we construct a sequence ( x n ) such that d ( x , x n ) < 1 / n and x n M \ U . Then x n x , so x is a limit point of M \ U . Problem 2 If ( M , d ) is a metric space, define the distance ( A , B ) of two nonempty sets A , B M as ( A , B ) = inf { d ( x , y ) : x A , y B } . Give an example of two closed subsets of R (with respect to the standard metric) such that ( A , B ) < d ( x , y ) for all x A , y B . Explain your answer....
View
Full
Document
 Fall '08
 RIEMAN
 Math

Click to edit the document details