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Wada basins

  1. Feb 22, 2010 #1
    What kind of sets exhibit the wada property? I know R2 does but does it extend to R3 or R itself?
     
  2. jcsd
  3. Feb 23, 2010 #2

    quasar987

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    The lakes (or basins) of Wada are an example in the plane of 3 disjoint sets sharing the same boundary. You get an analogous (n+2)-dimensional construction by "elevating" in the additional dimensions. Namely, if B,C,D are the 3 disjoint sets in R², take B x R^n, C x R^n, D x R^n, where x is the cartesian product.

    This takes care of every dimension except the first. In R, an example of 3 disjoint sets sharing the same boundary is A = {the points of the form 1/2n, n positive integer}, B= {the points of the form 1/(2n+1), n positive integer}, C={the points of the form -1/n, n positive integer}. Their common boundary being 0.
     
  4. Feb 23, 2010 #3
    Don't the sets have to be open??? Yes they do, also -1/n also has a boundary at -1, and 1/2n+1 and 1/2n have boundary points at 1/2, 1/3, so they don't have the same boundary.
     
  5. Feb 23, 2010 #4

    quasar987

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    Oh yes, you are right on all accounts. Plus, the sets have to be connected. So my exemple in R isn't good.

    Connected open sets in R are open intervals. Clearly, no three open intervals can have a common boundary: call I_1 the interval containing -[itex]\infty[/itex] and I_2 the interval containing +[itex]\infty[/itex]. Then if I_1=(-[itex]\infty[/itex],a) and I_2=(b,
    +[itex]\infty[/itex]), it must be that a=b if these two are to have the same boundary. But then I_1 u I_2 = R\{a} and so there is no place to put the third interval I_3.

    I wonder if it works if we remove the connectivity condition.
     
  6. Feb 23, 2010 #5
    I was trying to think of a topology that maybe the open sets have odd properties on R. I was told that maybe the complement of the cantor set would work , but I think that may be wrong too
     
  7. Feb 23, 2010 #6
  8. Feb 23, 2010 #7

    quasar987

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    Well, what odd properties are you after?
     
  9. Feb 23, 2010 #8
    Well I'm not sure.. Odd properties that make this problem easier? But it's kind of impossible to change the fact that Any open set will have an interval
     
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