$p$-adic metric on $\mathbb Q-\{0\}$-what are its interesting properties?

Question

Each non-zero rational number $x$ can be expressed as ${p^kr}\over {s}$ for a unique value of $k ∈ \mathbb Z$, where $r ∈ \mathbb Z$ and $s ∈\mathbb N$ and neither $r$ nor $s$ is divisible by $p$; we define $[x]_p$ to be $p^{−k}$.I think this is the $p$-adic metric.Does this metric have many interesting properties.I mean does it show behaviours that we do not find in $\mathbb R.$Doe it act as a counterexample to many problems?

Can someone help me to explore interesting facts about this metric?

Answer 1

It's an example of an ultrametric: a metric in which the triangle inequality can be strengthened to $$d(x,z) \le \max(d(x,y), d(y,z))$$

This has some interesting "geometric" consequences:

• all triangles are isosceles: if $x,y,z$ are distinct points of $X$, there are at most two different numbers in $\{d(x,y), d(x,z), d(y,z)\}$.

• every point of a ball is a centre of a ball: $a \in B(x,r) \implies B(x,r)=B(a,r)$.

• every open ball is closed: $B(x,r)=\overline{B(x,r)}$.

So all such spaces are zero-dimensional (implying there are no connected subsets except singletons). See Wikipedia for more links and info.