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Math Help - Metric Space

  1. #1
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    Metric Space

    I need to define a metric function "d" on the space C={0,1}^N (infinite {0,1} sequences)
    d must support:
    a. get only rational values
    b. for every c belongs to C and every \epsilon>0 the ball of radius \epsilon contains infinite number of objects.
    c. for every a,b belong to C there is an isometric function f : C -> C that supports f(a)=b.

    I apologize for any spelling mistakes and appreciate your help
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Consider the sequence space \sum_{2}=\left\{{s=(s_0,s_1,s_2,\ldots)}: s_i=0\;\vee\;s_i=1\right\} . Then, it is easy to prove that

    (i) d(s,t)=\displaystyle\sum_{i=0}^{+\infty}\dfrac{|s_  i-t_i|}{2^i} is a distance on \sum_{2}.

    (ii) If s,t\in\sum_2 and s_i=t_i for i=0,1,\ldots,n then, d(s,t)\leq 1/2^n . Conversely, if d(s,t)<1/2^n , then s_i=t_i for i\leq n .

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