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


by benorin
Tags: cantor, homeomorphic, integers, padic
benorin
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#1
Nov27-05, 11:34 AM
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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.
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matt grime
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#2
Nov27-05, 11:39 AM
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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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