331mt1soln

# 331mt1soln - Fall 2005 Math 331 Real Analysis First Midterm...

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Unformatted text preview: Fall 2005 Math 331 - Real Analysis First Midterm, 16/11/2005 1. Let x be a point in an arbitrary metric space ( M, d ) : (a) Show that if d is the discrete metric, then the subset { x } is both open and closed. Solution: Let A ⊂ M . If A = ∅ or M then A is both open and closed. Otherwise let x ∈ A . Since the open set B ( x, 1) = { x } lies in A , the set A is open by definition. Similarly, for y ∈ A c , the open ball B ( y, 1) = { y } is in A c hence the set A c is open and A is closed. (b) For d an arbitrary metric and ∈ R + , is X = B ( x, )- { x } open or closed or both or neither? ( Hint: solve part (a) first. ) Solution: If X is empty (which may happen for example in case of discrete metric) then X is both open and closed. If X 6 = ∅ then X is open because for each y ∈ X , Y = B ( y, min(- d ( x, y ) , d ( x, y )) lies entirely in X . An alternative proof for the fact that X is open is as follows: Since { x } is closed, { x } c is open. Observe that X = B ( x, )-{ x } = B ( x, ) ∩{ x } c ; i.e. X is the union of two open sets, so X is open....
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331mt1soln - Fall 2005 Math 331 Real Analysis First Midterm...

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