Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric

Throughout, $\mathbb{R}$ is the complete ordered field (Complete ordered field (least-upper-bound property), Ordered field) constructed in this library (The real numbers) and carrying its order (Order on the reals).

Let $X$ be a set. A metric on $X$ is a function $d : X \times X \to \mathbb{R}$ such that for all $x, y, z \in X$:

• (M1) Separation. $d(x,y) = 0$ if and only if $x = y$.
• (M2) Symmetry. $d(x,y) = d(y,x)$.
• (M3) Triangle inequality. $d(x,z) \le d(x,y) + d(y,z)$.

A metric space is a pair $(X,d)$ consisting of a set $X$ and a metric $d$ on it. The elements of $X$ are its points and $d(x,y)$ is the distance from $x$ to $y$. When only one metric is in play we write $X$ for $(X,d)$; when several are, the metric is always named.

The values of a metric are real numbers. The codomain is $\mathbb{R}$, so $d(x,y)$ is an honest element of the complete ordered field and every inequality above is an inequality there. No infinite value is permitted; Which metric axiom list this library uses, the live naming fork between semimetric and pseudometric, and why extended metrics are not treated here records why extended metrics are not treated in this library.

Nonnegativity is deliberately absent from the axiom list. Many texts add a fourth axiom $d(x,y) \ge 0$. It is redundant: (M1), (M2) and (M3) already force it, as Nonnegativity of a metric is a consequence of the other axioms, not an axiom proves. Nothing below assumes it before that lemma is available.

Pseudometric. A pseudometric on $X$ is a function $p : X \times X \to \mathbb{R}$ satisfying (M2), (M3) and the weakening

• (M1') Reflexivity. $p(x,x) = 0$ for every $x \in X$

of (M1). A pseudometric may therefore assign distance $0$ to two distinct points. Every metric is a pseudometric, and a pseudometric is a metric exactly when $p(x,y) = 0$ forces $x = y$.

Ultrametric. An ultrametric on $X$ is a metric $d$ that in addition satisfies

• (M3') Strong triangle inequality. $d(x,z) \le \max\{d(x,y), d(y,z)\}$

for all $x, y, z \in X$, where the maximum is that of a two-element subset of $\mathbb{R}$, which exists and is one of the two elements (Maximum and minimum of a set, Every nonempty finite set of reals has a maximum and a minimum). An ultrametric space is a pair $(X,d)$ with $d$ an ultrametric.

• (M3') is a genuine strengthening of (M3), not an independent axiom on top of it. A function satisfying (M1), (M2) and (M3') automatically satisfies (M3): by Nonnegativity of a metric is a consequence of the other axioms, not an axiom such a function is nonnegative, and for nonnegative reals $a, b$ one has $\max\{a,b\} \le a + b$, since the maximum is one of $a, b$ and the other summand is $\ge 0$. So "a metric satisfying (M3')" and "a function satisfying (M1), (M2), (M3')" describe the same objects, and the definition above may be read either way.

• Why the biconditional form of (M1). Splitting (M1) into "$d(x,x) = 0$" and "$d(x,y) = 0 \Rightarrow x = y$" gives the same notion; the split form is what makes the pseudometric weakening above a matter of deleting one clause. The naming fork between pseudometric and semimetric, which is live in the literature, is settled for this library in Which metric axiom list this library uses, the live naming fork between semimetric and pseudometric, and why extended metrics are not treated here.

• The metric is part of the data. Two different metrics on the same set are two different metric spaces, even when they have the same open sets. That is why Topologically, uniformly and Lipschitz equivalent metrics on a set compares metrics at three separate strengths rather than one, and why a property can be invariant under one of them and not under another (FALSE: boundedness of a metric space is determined by its topology).