The post-office metric $d(x,y)=\mathrm{\parallel }x\mathrm{\parallel }+\mathrm{\parallel }y\mathrm{\parallel }$ for $x\mathrm{\ne }y$ on ${\mathbb{R}}^n$, and its isolated points

Let $n\ge 1$ and let ${\mathbb{R}}^n$ carry the Euclidean metric $d_2$ of ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it. Write $\mathbf{0}$ for the element of ${\mathbb{R}}^n$ with all coordinates $0$ and

$\mathrm{\parallel }x\mathrm{\parallel }:=d_2(x,\mathbf{0})=\sqrt{{\sum }_{k<n}{x}_k^2}\mathrm{.}$

Define the post-office metric (also called the SNCF metric) by

$\rho (x,y):=\{\begin{array}{ll}0& x=y,\\ \mathrm{\parallel }x\mathrm{\parallel }+\mathrm{\parallel }y\mathrm{\parallel }& x\mathrm{\ne }y\mathrm{.}\end{array}$

The name is the picture: to travel between two places you must first go to the central post office at $\mathbf{0}$. Then:

1. $\rho$ is a metric on ${\mathbb{R}}^n$ (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric).
2. Every $x\mathrm{\ne }\mathbf{0}$ is an isolated point of $({\mathbb{R}}^n,\rho )$ (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space), because $B_{\rho }(x,\mathrm{\parallel }x\mathrm{\parallel })=\{x\}$.
3. $\mathbf{0}$ is not an isolated point: every ball $B_{\rho }(\mathbf{0},r)$ contains points other than $\mathbf{0}$.

So $({\mathbb{R}}^n,\rho )$ has exactly one non-isolated point, which no metric equivalent to $d_2$ could achieve: in the Euclidean topology no point of ${\mathbb{R}}^n$ is isolated.

• The topology is almost discrete. Every singleton $\{x\}$ with $x\mathrm{\ne }\mathbf{0}$ is open by claim 2, so every subset of ${\mathbb{R}}^n\setminus \{\mathbf{0}\}$ is open (Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed); the only points a set has to be careful about are those near $\mathbf{0}$.
• This is not equivalent to any of $d_1$, $d_2$, $d_{\mathrm{\infty }}$, not even topologically. Those three share a topology (The metrics $d_1$, $d_2$ and $d_{\mathrm{\infty }}$ on ${\mathbb{R}}^n$ are metrics and are Lipschitz equivalent, with explicit constants) in which no point is isolated: for $x\in {\mathbb{R}}^n$ and $r>0$ the element $y$ with $y_0:=x_0+r\mathrm{/}2$ and $y_k:=x_k$ for $1\le k<n$ satisfies $d_2(x,y)=r\mathrm{/}2<r$, by the computation of step 1.3, and differs from $x$. In $\rho$, by claim 2, every point but $\mathbf{0}$ is isolated.
• The same construction works over any metric space with a distinguished point, replacing $\mathrm{\parallel }x\mathrm{\parallel }$ by the distance to that point; nothing above uses more about $\mathrm{\parallel }\cdot \mathrm{\parallel }$ than nonnegativity and vanishing exactly at $\mathbf{0}$.