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Math Help - p-adic metric

  1. #1
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    p-adic metric

    I have the p-adic metric on Q. I need to show that p-adic and q-adic metrics are not equivalent for distinct primes p and q. How can I go about this?
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  2. #2
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    exhibit a sequence that converges in one but not the other.
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  3. #3
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    Does (p/q), (p/q)^2, (p/q)^3, ... work? As clearly if this is (x_n) then I have d_p(x_n, 0) = p^{-n} -> 0, and d_q(x_n,0) = q^{n} which does not converge.

    What is the reasoning now, if the above is correct, behind this being a reason for the two metrics to be non-equivalent? I apologise for the rather ignorant questioning.

    Edit: I am assuming that this would mean that the identity map is discontinuous in at least one direction, which would be the reason for the non-equivalence; however I am interested here.
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  4. #4
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    Quote Originally Posted by ihateyouall View Post
    Edit: I am assuming that this would mean that the identity map is discontinuous in at least one direction, which would be the reason for the non-equivalence; however I am interested here.

    correct, the identity map is discontinuous so the two metrics are not inducing the same topology so they cannot be equivalent.
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