pseudometrics

A pseudometric on a set X is a function d: X × X → [0, ∞) that satisfies three properties for all x, y, z in X: symmetry d(x, y) = d(y, x), nonnegativity d(x, y) ≥ 0, and the triangle inequality d(x, z) ≤ d(x, y) + d(y, z). In addition, a pseudometric requires that d(x, x) = 0 for every x. What distinguishes a pseudometric from a metric is the relaxation of the identity of indiscernibles: a pseudometric may have d(x, y) = 0 for x ≠ y.

A pseudometric induces a natural topology on X and makes X into a pseudometric space. If d(x, y) = 0 is possible for distinct points, the space is not necessarily Hausdorff. However, by identifying points that are at zero distance, one can form a quotient space X/~ where x ~ y iff d(x, y) = 0. The quotient is a metric space with distance d([x], [y]) = d(x, y).

Common constructions give rise to many pseudometrics. For any function f: X → R, the quantity d(x, y) = |f(x) − f(y)| is a pseudometric; it is a metric exactly when f is injective. More generally, if p is a seminorm on a vector space V, then d(u, v) = p(u − v) is a pseudometric, since p may vanish on nonzero vectors.

Pseudometrics are widely used to induce topologies, measure similarity, and define distances when the identity of indiscernibles is too strong a requirement. They appear in analysis, functional analysis, and data-oriented contexts where distinct objects may be indistinguishable under a given criterion.