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Question about isolated points.

  1. Mar 18, 2012 #1
    1. The problem statement, all variables and given/known data
    If I just had the set containing [itex] \pi [/itex] on the real line.
    So this is an isolated point. Is this set closed?
    3. The attempt at a solution
    I think this set is closed because it contains its limit points, because it only has one point.
    Am i thinking about this correctly?
     
  2. jcsd
  3. Mar 18, 2012 #2
    Hmm, that's an interesting question. I guess it would be closed because that's the only point in the set, and there's no option for an e-neighbourhood around it.
     
  4. Mar 18, 2012 #3

    Dick

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    I think you are thinking about it correctly but that you are even asking this question makes me wonder. Can you explain your doubts further? What's the exact definition of a limit point?
     
  5. Mar 18, 2012 #4
    thats what I thought too
     
  6. Mar 18, 2012 #5
    If we wanted to put things on more definite footing, we could assume that it isn't closed. Then there is some point x1 such that for all E > 0 there is some x in our set such that d(x,x1) < E. However, since x = π in all cases, then the only such point is π, a contradiction.
     
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