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Question about Dense sets in R.

  1. Apr 26, 2012 #1
    1. The problem statement, all variables and given/known data
    Decide wheter the following sets are dense in ℝ, nowwhere dense in ℝ
    , or somewhere in between.
    a) A= [itex] \mathbb{Q} \bigcap [0,5] [/itex]
    b) [itex] B= \{ \frac{1}{n} : n \in \mathbb{N} [/itex]
    d) the cantor set.
    3. The attempt at a solution
    a) so we have the rationals intersected with [0,5], so we have all the rationals from
    [0,5], which is dense on that interval.
    b) this set does not seem like it is dense, except maybe close to zero, but even then
    I could find an interval close to zero that did not contain any elements in my set.
    d) The cantor set is nowhere dense because it has no intervals, if I have an interval of positive length stating at one of the endpoints of the cantor set it would stick out and it wouldn't contain any points in my set.
     
  2. jcsd
  3. Apr 26, 2012 #2

    Office_Shredder

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    Your answer for d is confusing, a set does not need to contain an interval to be dense, for example the rationals
     
  4. Apr 26, 2012 #3
    is the second thing I said in part d okay. " if I have an interval of positive length stating at one of the endpoints of the cantor set it would stick out and it wouldn't contain any points in my set."
     
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