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