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Thread: Metric Space

  1. #1
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    Metric Space

    I need to define a metric function "d" on the space $\displaystyle C={0,1}^N$ (infinite {0,1} sequences)
    d must support:
    a. get only rational values
    b. for every c belongs to C and every $\displaystyle \epsilon>0$ the ball of radius $\displaystyle \epsilon$ contains infinite number of objects.
    c. for every a,b belong to C there is an isometric function f : $\displaystyle C -> C$ 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 $\displaystyle \sum_{2}=\left\{{s=(s_0,s_1,s_2,\ldots)}: s_i=0\;\vee\;s_i=1\right\}$ . Then, it is easy to prove that

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

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

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