The $p$-adic metric is defined on $\mathbb Q$ as follows:
$\rho(x,y)=p^{-\alpha}$ where $x-y={r\over s}p^{\alpha},gcd(r,s)=1,p\ $ is prime.
Now, if I take $x=10,y=4$ then $x-y=6=3.2$ Then what is $\rho (x,y)?$ By definition, both $1\over 2$ and $1\over 3$ qualify.
Is it fine for a metric to to give different values for the same two points?Don't the other metrics on $\mathbb R$ or $\mathbb Q$ give unique value for same points.