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

Example

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 $d_1$ , $d_2$ , $d_\infty$ are metrics on it. Write $\mathbf{0}$ for the element of $\mathbb{R}^n$ with all coordinates $0$ and

$\|x\| := d_2(x, \mathbf{0}) = \sqrt{\textstyle\sum_{k<n} x_k^2} .$

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

$\rho(x,y) := \begin{cases} 0 & x = y, \\ \|x\| + \|y\| & x \ne y. \end{cases}$

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

$\rho$ is a metric on $\mathbb{R}^n$ (Metric space: $d(x,y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric).
Every $x \ne \mathbf{0}$ is an isolated point of $(\mathbb{R}^n, \rho)$ (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space), because $B_\rho\big(x, \|x\|\big) = \{x\}$ .
$\mathbf{0}$ 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.

Facts & Assumptions

Given: A natural $n \ge 1$ ; elements $x, y, z \in \mathbb{R}^n$ ; a real $r > 0$ ; and the element $w \in \mathbb{R}^n$ with $w_0 := r/2$ and $w_k := 0$ for $1 \le k < n$ .

[L1]

$d_2$ is a metric on $\mathbb{R}^n$ ( $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$ , and $d_1$ , $d_2$ , $d_\infty$ are metrics on it, Metric space: $d(x,y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric), so $\|x\| = d_2(x,\mathbf{0}) \ge 0$ (Nonnegativity of a metric is a consequence of the other axioms, not an axiom) and $\|x\| = 0$ exactly when $x = \mathbf{0}$ .

[L2]

Finite sums (Laws of finite sums and finite products, Finite sums and finite products, by recursion): splitting a sum at an index $m \le n$ , and $\sum_{k<p}\lambda = p\lambda$ , so a sum all of whose terms are $0$ is $0$ ; and $\mathbb{N}$ contains $0$ , so the index $0$ exists as soon as $n \ge 1$ (The natural numbers $\mathbb{N}$ (von Neumann)).

[L3]

Square roots: $\sqrt{t^2} = t$ for $t \ge 0$ (Square roots exist: a unique $\sqrt{a} \ge 0$ with $(\sqrt{a})^2 = a$ ; the positives are $\{x^2 : x \neq 0\}$ ).

[L4]

Halving: for a real $r > 0$ the element $r/2 := r(1+1)^{-1}$ is positive with $r/2 + r/2 = r$ , hence $r/2 < r$ ; this uses $0 < 1$ , positivity of inverses of positives, and positivity of products of positives (The multiplicative identity is positive, Order is preserved by adding a constant and by adding inequalities, Inverses of positives are positive, and reciprocation reverses order, Sign rules for products and monotonicity of multiplication, Field, Ordered field).

[L5]

Order arithmetic: inequalities may be added, a nonnegative term may be dropped from the larger side, and by trichotomy $a \ge b$ rules out $a < b$ (Order is preserved by adding a constant and by adding inequalities, Ordered field, Complete ordered field (least-upper-bound property)).

[L6]

Balls, isolated points: $B_\rho(u,t) = \{v : \rho(u,v) < t\}$ , and $u$ is isolated in a set $A \ni u$ when some ball meets $A$ only in $u$ (Open ball, closed ball and sphere in a metric space, Interior, closure, boundary, limit point, isolated point and dense subset of a metric space).

Verification

technique · direct

1.1

Basic values: $\rho(u,v) \ge 0$ for all $u,v$ , being $0$ or a sum of two nonnegative numbers; $\rho$ is symmetric, since both defining clauses are; and $\rho(u,v) = 0$ exactly when $u = v$ , because for $u \ne v$ the value $\|u\| + \|v\|$ vanishes only if $\|u\| = \|v\| = 0$ , that is only if $u = v = \mathbf{0}$ , contradicting $u \ne v$ .

L1L5

1.2

Triangle inequality: if $x = z$ then $\rho(x,z) = 0 \le \rho(x,y) + \rho(y,z)$ ; if $x \ne z$ and $y = x$ , then $y \ne z$ and $\rho(x,y) + \rho(y,z) = 0 + \|x\| + \|z\| = \rho(x,z)$ ; if $x \ne z$ and $y = z$ , the same computation applies with the roles exchanged; and if $x \ne z$ with $y \ne x$ and $y \ne z$ , then $\rho(x,y) + \rho(y,z) = \|x\| + \|y\| + \|y\| + \|z\| \ge \|x\| + \|z\| = \rho(x,z)$ , since $\|y\| \ge 0$ .

L1L5

1.3

The element $w$ satisfies $\sum_{k<n} w_k^2 = w_0^2 + 0 = (r/2)^2$ , splitting the sum at index $1 \le n$ and using that all remaining terms are $0$ ; hence $\|w\| = \sqrt{(r/2)^2} = r/2 < r$ , and $w \ne \mathbf{0}$ because $w_0 = r/2 \ne 0$ .

L2L3L4

2.1

Claim 1: $\rho$ satisfies (M1) and (M2) by step 1.1 and (M3) by step 1.2, so it is a metric on $\mathbb{R}^n$ .

step 1.1step 1.2

3.1

Claim 2: let $x \ne \mathbf{0}$ , so $\|x\| > 0$ by [L1] and trichotomy. For $y \ne x$ we get $\rho(x,y) = \|x\| + \|y\| \ge \|x\|$ , so $y \notin B_\rho(x,\|x\|)$ ; and $x \in B_\rho(x,\|x\|)$ since $\rho(x,x) = 0 < \|x\|$ . Hence $B_\rho(x,\|x\|) = \{x\}$ and $x$ is isolated in $\mathbb{R}^n$ .

step 1.1step 2.1L1L5L6

3.2

Claim 3: let $r > 0$ . For $y \ne \mathbf{0}$ one has $\rho(\mathbf{0},y) = \|\mathbf{0}\| + \|y\| = \|y\|$ , so the element $w$ of step 1.3 satisfies $\rho(\mathbf{0},w) = \|w\| < r$ and $w \ne \mathbf{0}$ ; thus $B_\rho(\mathbf{0},r)$ contains a point other than $\mathbf{0}$ , for every $r > 0$ , and $\mathbf{0}$ is not isolated.

step 1.3step 2.1L1L6

4.1

Claims 1, 2 and 3 hold by steps 2.1, 3.1 and 3.2.

step 2.1step 3.1step 3.2∎

Remarks

The topology is almost discrete. Every singleton $\{x\}$ with $x \ne \mathbf{0}$ 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 $d_1$ , $d_2$ , $d_\infty$ , not even topologically. Those three share a topology (The metrics $d_1$ , $d_2$ and $d_\infty$ 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 $y_0 := x_0 + r/2$ and $y_k := x_k$ for $1 \le k < n$ satisfies $d_2(x, y) = r/2 < r$ , 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 $\|x\|$ by the distance to that point; nothing above uses more about $\|\cdot\|$ than nonnegativity and vanishing exactly at $\mathbf{0}$ .

Depends on

Used by

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Direct dependencies and their dependencies through the next three levels: 70 results over 25 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Metric space (Wikipedia) (standard reference, not scraped)
Isolated point (Wikipedia) (standard reference, not scraped)
Euclidean space (Wikipedia) (standard reference, not scraped)
SNCF metric (PlanetMath) (standard reference, not scraped)

The post-office metric d(x,y)=∥x∥+∥y∥d(x,y) = \|x\| + \|y\|d(x,y)=∥x∥+∥y∥ for x≠yx \ne yx=y on Rn\mathbb{R}^nRn, and its isolated points