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There is a point p not in E s.t for any e>0

  1. Feb 1, 2010 #1
    Let E be a proper subset of R. There is a point p not in E s.t for any e>0, there exists a point q in E s.t |p-q|<e. Prove that E is not compact.

    Proof:

    p is in R-E. For a e>0, p+e is in E. So R-E is closed on one side which implies E is open on one side. By using heine-borel thrm we can conclude that E is not compact.


    Is this proof valid?
     
  2. jcsd
  3. Feb 2, 2010 #2
    Re: Proof

    Let N(p) be a neighbourhood of p. What is N(p) intersect E? Does it intersect E at any point other than p? Use this and what you know about closed sets/limit points.
     
  4. Feb 2, 2010 #3
    Re: Proof

    From what you are saying, it seems that p is a boundary point of E. But p doesn't belong to E. Now, for a set to be compact in R, it must be what?
     
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