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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?

User
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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.

Joonas Ilmavirta
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  • What do you mean that "this is false for the other two spaces"? – User Aug 18 '14 at 18:00
  • In $\mathbb Q$ and $\mathbb Z$ there is no element with the property that no two distinct elements have the same distance to it. – Joonas Ilmavirta Aug 18 '14 at 18:01
  • I guess I don't understand what you mean by "no two distince elements have the same distance to it." could you explain that? – User Aug 18 '14 at 18:03
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    In $\mathbb{N}$ you can only be "on the right" of $0$. That's what he's talking about. – bartgol Aug 18 '14 at 18:06
  • @Yk26, there is no point $x\in\mathbb Z$ with this property: there are no two points $a,b$ (distinct from $x$ and each other) so that $|x-a|=|x-b|$. – Joonas Ilmavirta Aug 18 '14 at 18:07
  • I'm still confused. For example, I'm thinking $x=0$ and $a=-1$ and $b=1$ in the above explanation. – User Aug 18 '14 at 18:13
  • @Yk26, if you are in $\mathbb Z$, then for any $x$ the points $a=x-1$ and $b=x+1$ have the same distance to $x$. But if you are in $\mathbb N$ and $x=0$, you can't find any such $a$ and $b$. – Joonas Ilmavirta Aug 18 '14 at 18:17