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Proof nowhere-dense of a closure's complement

  1. Nov 12, 2011 #1
    1. The problem statement, all variables and given/known data
    A set E in ℝ is nowhere dense if and only if the closure of E's complement(E with a line over it) is dense in ℝ


    2. Relevant equations
    I need help proving this lemma. I'm not entirely sure where to start.


    3. The attempt at a solution
    I know we have to proof it two ways, backwards and forwards. For the forward, all I can think to use if the fact that the closure of E complement contains no nonempty open intervals, but I don't know where to go with that.

    Any help would be appreciated.
     
  2. jcsd
  3. Nov 12, 2011 #2

    micromass

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    First, how did you define "nowhere dense"??

    Second, do you know the following formula's:

    [tex]cl(X\setminus E)=X\setminus int(E)~\text{and}~int(X\setminus E) = X\setminus cl(E)[/tex]

    These equalities will prove to be handy. Try to prove them!!

    Note: cl means closure and int means interior.
     
  4. Nov 12, 2011 #3
    micromass-

    I have never seen that equation before.

    I have nowhere dense defined as: a set E in ℝ is nowhere dense if the closure of E contains no non-empty intervals.

    I will look at those equations and see what I can do. Thank you!
     
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