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Homework Help: Equivalent metrics

  1. Oct 8, 2008 #1
    1. The problem statement, all variables and given/known data
    Show that d and p are equivalent metrics on X where p=d(x,y)/(1+d(x,y))



    2. Relevant equations
    ive proved already that p is indeed a metric too (if d is a metric).


    3. The attempt at a solution

    I believe im supposed to use the Lipschitz condition where there exits constants A and B st for all x,y,

    Ap<=d<=Bp

    but i think i can actually prove that one of these two constants cannot exist... and i using the wrong definitions? Thanks!
     
  2. jcsd
  3. Oct 8, 2008 #2

    Dick

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    Homework Helper

    Two metrics are equivalent if they induce the same topology. Isn't that the real definition? It is true that Ap<=d<=Bp shows that. But isn't it also true that for a metric topology what really determines the topology is the 'small' sets? If you can show there is a condition like that for say d<=1. That would also suffice, it doesn't have to hold for ALL x,y. Just for nearby ones.
     
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