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Transfinite donuts?

  1. Apr 28, 2010 #1
    Has anyone ever developed any sort of math involving donuts with an infinite number of holes? By donut, I mean a two-dimensional closed surface, curved in 3-space, with one 'hole'. Are there any results, of any kind, for 2-D donuts in 3-D space, with infinite number of holes?
     
  2. jcsd
  3. Apr 28, 2010 #2
    I think they're about as well understood as the donut with one hole. Riemann surfaces are one of the most thoroughly understood branches of mathematics.
     
  4. Apr 30, 2010 #3

    lavinia

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    Could you construct a doughnut with uncountable many holes ? It would not be a paracompact manifold.
     
  5. Apr 30, 2010 #4
    You mean something like [tex]S\times I[/tex], where S is the unit disk with the rational points inside a circle of radius 1/2 centered at the origin deleted?
     
    Last edited: Apr 30, 2010
  6. Apr 30, 2010 #5

    lavinia

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    Not sure how that example is a torus.

    I was thinking more of the long line Cartesian product the circle with uncountably many holes removed.
     
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