Pseudodistances

A pseudodistance, commonly called a pseudometric, is a function d defined on a set X that measures a notion of distance between pairs of elements. It satisfies non-negativity, symmetry, and the triangle inequality, and it assigns zero distance to every element with itself (d(x, x) = 0 for all x). Unlike a metric, a pseudodistance need not distinguish distinct points, meaning d(x, y) = 0 may occur even when x ≠ y.

Formally, a pseudodistance d: X × X → [0, ∞) satisfies:

- d(x, y) ≥ 0 for all x, y in X

- d(x, y) = d(y, x) for all x, y

- d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z

- d(x, x) = 0 for all x

The potential failure of the identity of indiscernibles leads to the notion of an equivalence relation: x is related to y if d(x, y) = 0. The quotient X/~, where ~ is this relation, inherits a metric structure from d, turning zero-distance classes into single points.

Constructions and examples:

- Seminorms: If p is a seminorm on a vector space V, then d(u, v) = p(u − v) is a pseudometric. The kernel of p (points with zero seminorm distance) yields nontrivial pairs with distance zero.

- Equivalence-based pseudometrics: Given any equivalence relation ~ on X, define d(x, y) = 0 if x ~ y, and d(x, y) = 1 otherwise. This d is a pseudometric, capturing the partition into equivalence classes.

- Quotients and topologies: Pseudodistances often generate topologies that reflect a coarse separation of points, with completeness and convergence interpreted up to zero-distance.

In practice, pseudodistances are used to study spaces where a genuine metric is too rigid, providing a framework for analysis, topology, and the formation of quotient metric spaces. A pseudodistance becomes a metric precisely when d(x, y) = 0 implies x = y.