hw4keykey.pdf - Introductory Real Analysis Math 327 Winter...

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Introductory Real Analysis University of Washington Math 327, Winter 2018 c 2018, Dr. F. Dos Reis Homework 4 Exercise 1. If S n is a sequence of open sets, and C n a sequence of open sets, 1. Prove that [ n =1 S n is an open set. Key: Let S be [ n =1 S n . Let’s prove the S is open by checking that it satisfies the definition of open sets. For any x in S , there exists N such the x S N . S N is an open set and x is an element of S N . By definition of open set, there exists a neighborhood ( x - h N , x + h N ) entirely included in S N . Since S N S , the neighborhood ( x - h N , x + h N ) is entirely included in S and S is open. 2. Prove that the finite intersection N \ n =1 S n is open. Key: Let T = N \ n =1 S n . For any x in T . By definition of the intersection x S k for any k ∈ { 1 , 2 , · · · , N } . Each set S k is open, therefore there exists a neighborhood ( x - h k , x + h k ) entirely included in S k . The set { h k , k ∈ { 1 , · · · , N }} is a finite sets, therefore it has a smallest element h min . Since each h k is positive, h min is positive. For any k ∈ { 1 , · · · , N } , the neighborhood ( x - h min , x + h min ) ( x - h k , x + h k ) S k Therefore there exists a neighborhood, ( x - h min , x + h min ) entirely included in T . 3. Is \ n =1 S n an open set? Prove the result or find a counterexample. Key: Cant we use the same proof as before for an infinite intersection? What works: for any x in \ n =1 S n , x is in S k for any k N . S k is open therefore there exists a neighborhood ( x - h k , x + h k ) entirely included in S k . Consider the set { h k } . It is now any infinite set of positive numbers. It may not necessarily have a smallest element but it has a least upper bound h . If the least upper bound is positive then h 6 h k for any kin N and ( x - h, x + h ) ( x - - h k , x + h k ) S k for any k N and \ n =1 S n is open. The problem comes if h = 0. in this case, there is no neighborhood included in every S n and the proof fails.
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  • Fall '08
  • Topology, Empty set, Metric space, Sn, Closed set

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