Give an example of a metric space not homeomorphic to any subspace of Euclidean space $\Bbb{R}^n$

Question

The question is exactly like in the title.

Answer 1

If I understand correctly, you are asking for a metric space $(\Bbb{R}^n, d)$ which is not homeomorphic to $(\Bbb{R}^n, d_2)$ where $d_2$ is the standard Euclidean metric. In this case, let $d$ be the discrete metric. Then $(\Bbb{R}^n, d)$ is not homeomorphic to $(\Bbb{R}^n, d_2)$. This is because we can find non-open sets in $(\Bbb{R}^n, d_2)$ but every set is open in $(\Bbb{R}^n, d)$, hence there are no non-open sets, so the two cannot be homeomorphic. The same discussion but for $\Bbb{R}$ can be found here.