1. ## 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

2. 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 $\displaystyle i\leq n$ .

Try the rest of your questions.