Which metric axiom list this library uses, the live naming fork between semimetric and pseudometric, and why extended metrics are not treated here

The axiom list. Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric asks a metric $d : X \times X \to \mathbb{R}$ for exactly three things: (M1) $d(x,y) = 0$ if and only if $x = y$; (M2) $d(x,y) = d(y,x)$; (M3) $d(x,z) \le d(x,y) + d(y,z)$. Many texts add a fourth, $d(x,y) \ge 0$, or build it into the codomain by writing $d : X \times X \to [0,\infty)$. That fourth condition is redundant: it follows from the other three, and Nonnegativity of a metric is a consequence of the other axioms, not an axiom proves it. The list is kept minimal here so that every verification of "is this a metric" has three things to check and not four, and so that no proof can quietly assume nonnegativity before it has been established.

Splitting (M1). Some texts state (M1) as two conditions, $d(x,x) = 0$ for all $x$ together with the implication $d(x,y) = 0 \Rightarrow x = y$. That is the same notion, and the split form is convenient because deleting the second half is exactly the weakening that produces a pseudometric.

The naming fork, which is live and is why this library says pseudometric. Two different weakenings of the axiom list circulate under overlapping names.

• Dropping the implication $d(x,y) = 0 \Rightarrow x = y$, keeping symmetry and the triangle inequality, gives what most current sources, and this library, call a pseudometric (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric). This is the notion a seminorm induces.
• Dropping the triangle inequality, keeping (M1) and (M2), gives what most current sources call a semimetric.

The fork is that a substantial part of the literature, especially in functional analysis and in older texts, uses semimetric for the first of these, that is as a synonym for pseudometric. There is no way to use the word semimetric here without inheriting the ambiguity, so this library does not use it at all: the first weakening is always called a pseudometric, and the second, which nothing here needs, is never named. Dropping symmetry instead gives a quasimetric, also not treated here; note that Nonnegativity of a metric is a consequence of the other axioms, not an axiom uses symmetry, so a quasimetric is not automatically nonnegative and the fourth axiom is not redundant for it.

Ultrametrics. The strong triangle inequality $d(x,z) \le \max\{d(x,y), d(y,z)\}$ implies (M3) in the presence of (M1) and (M2), by Nonnegativity of a metric is a consequence of the other axioms, not an axiom and the fact that the maximum of two nonnegative reals is at most their sum. So an ultrametric is a metric, and the definition may be read either as "a metric that also satisfies (M3')" or as "a function satisfying (M1), (M2) and (M3')". The two readings pick out the same objects.

Why extended metrics are not treated here. An extended metric is allowed to take the value $+\infty$, so that its codomain is $[0,\infty]$ rather than $[0,\infty)$; the axioms are read with the usual arithmetic of $+\infty$. The construction is useful, for instance when one wants to glue metric spaces without connecting them, and it is standard in metric geometry. It is not treated here, for one reason: its values would have to live in the extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$, whereas the axioms of Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric are stated over the complete ordered field $\mathbb{R}$ (Complete ordered field (least-upper-bound property)) and are never read anywhere else. Why they are kept there is set out in Conventions: $\sup \emptyset$, unbounded sets, and the extended reals: $\overline{\mathbb{R}}$ is not a field, the expressions $(+\infty) + (-\infty)$ and $0 \cdot (+\infty)$ have no definition compatible with the field axioms, and writing an infinite value silently moves the discussion into a different structure, after which every algebraic step needs its own justification. Every value of every metric in this library is therefore an element of $\mathbb{R}$.

Two consequences of that decision are visible on this page and are not oversights. First, an unbounded set has no diameter at all here, rather than a diameter $+\infty$ (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space). Second, the supremum metric is defined on the bounded real-valued functions only (The supremum metric $d_\infty(f,g) = \sup_x |f(x) - g(x)|$ is a metric on the bounded real-valued functions on a nonempty set), where texts working in $\overline{\mathbb{R}}$ define it on all of them.

Adding extended metrics honestly would mean restating Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric over a totally ordered set with a greatest element, carrying its own partial arithmetic, and re-proving over it everything this page proves over $\mathbb{R}$. No such restatement is made anywhere in this library, and until one is, every metric here takes real values.