As per the definition of metric space (Rudin): 
[![Rudin's definition of metric spaces](https://i.stack.imgur.com/O3lo6.png)](https://i.stack.imgur.com/O3lo6.png)
Here is what I understand: Any countable set $A$ can be mapped 1-1 to the set of natural numbers. Let $n(x)$ be the natural number to which any element $x$ belonging to $A$ is mapped. Define the distance function, $ d(p,q) = |n(p) - n(q)|$ for any element $p$ and $q$ belonging to A. This satisfies conditions $ (a),(b)$ and $(c) $. Therefore any countable set should be a metric space. 

Is there another way to reason this?