Metrics on $\mathbb{N}$ such that $(\mathbb{N}, d)$ and $(\mathbb{N}, d')$ are not homeomorphic

Question

I want to find two metrics $d$ and $d'$ on $\mathbb{N}$ such that $(\mathbb{N}, d)$ and $(\mathbb{N}, d')$ are not homeomorphic, but I'm having trouble doing so. I've tried using the discrete metric as this often has very different properties to the usual metric, and can be used to show it for $\mathbb{R}$ but I can't seem to get it to work for $\mathbb{N}$. I've also tried considering topological properties such as compactness, connectedness, etc, but again the two metrics always seem to have the same properties, and also considered sequences.

Any hints would be greatly appreciated!

Answer 1

A concrete example of Ethan's idea:

$d_1(n,m)=1$ for $n \neq m$ (the discrete metric), which has the property that any convergent sequence is eventually constant.

$d_2(0,n)=\frac{1}{n}$ for $n \neq 0$ and $d_2(n,m)= |\frac{1}{n}-\frac{1}{m}|$ for $n,m$ non-zero and distinct. Then $x_n = n$ converges to $0$, and such a non-trivial sequence does not exist under $d_1$, so the topologies are different (also $(\mathbb{N}, d_2)$ is compact, while $(\mathbb{N}, d_1)$ is not).

Answer 2

Let $A$ be a countable subset of any metric space you like. Transfer the metric to $\mathbb{N}$ with a bijection that establishes the countability.