What is the difference between $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ in terms of their metrics?

Question

what is the difference between $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ in terms of their metics?

Do I need more assumptions to make difference between them beside just their metric functions?

Answer 1

If you look at $\mathbb Q$, $\mathbb Z$ and $\mathbb N$ as metric spaces disregarding the ambient real axis, you can distinguish them. The space $\mathbb Q$ is the only one of them without isolated points (the others have only isolated points). The space $\mathbb N$ contains such an element that no two other elements have the same distance to it; this is false for the other two spaces.