Set of p-adic integers is homeomorphic to Cantor set; how?

  1. benorin

    benorin 1,026
    Homework Helper

    Could somebody explain with due brevity why/how the set of p-adic integers is homeomorphic to the Cantor set less one point for any prime p?

    This is a quote from Wikipedia:Cantor Set: "The Cantor set is also homeomorphic to the p-adic integers, and, if one point is removed from it, to the p-adic numbers."

    Can somebody explain this simply, I don'y really get p-adic #'s.

    P.S. Not homework, don't want a proof, just understanding of it.
  2. jcsd
  3. matt grime

    matt grime 9,395
    Science Advisor
    Homework Helper

    Write down the bijection from the traditional representation of the cantor set as the reals in [0,1] with no 1's in the base three expansion to the 2-adics (write backwards and put 1s instead of 2s at all points), it is not a deep topological property we're talking about, just a formal one, a little like the integers with the discrete topology are homeomorphic to the positive integers with the discrete topology.
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