Closeness in p-adic numbers
Two p-adic numbers are close if they are both divisible by $p^n$ for $n$ large. From the page:
Remark. Consider the absolute value $|x-y|=s^{-v_p(x-y)}$ described above. Intuitively, $x$ and $y$ are close if their difference is close to $0$. Since a coordinate of $x-y$ being 0 means automatically that all lower coordinates are 0, we see that $x-y$ become closer if they share the first $n$ coordinates, where $n$ is large. But this means that $v_p(x-y)$ gets bigger and bigger. So for this intuitive closeness to be reflected in the absolute value, we want to raise $s$ to the negative power, so that a larger $v_p(x-y)$ results in a smaller distance between $x$ and $y$.