Metric Space

**aharonidan** · February 27th 2011, 12:58 PM

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

**FernandoRevilla** · February 28th 2011, 08:53 AM

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 $i\leq n$ .

Try the rest of your questions.

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