quasimetrics

Quasimetrics are a generalization of metric spaces in which the distance function may be non-symmetric. Formally, a quasi-metric space consists of a set X together with a function d: X×X → [0, ∞) such that for all x, y, z in X: d(x, x) = 0 and d(x, z) ≤ d(x, y) + d(y, z). Unlike a metric, a quasi-metric is not required to satisfy d(x, y) = d(y, x). The lack of symmetry means that the distance from x to y can differ from the distance from y to x.

There are related notions that refine or loosen the axioms further. A quasi-pseudometric relaxes the identity of indiscernibles, allowing d(x, y) = 0 for x ≠ y. In practice, many authors use “quasi-metric” to refer to the non-symmetric, triangle-inequality version, while “quasi-pseudometric” emphasizes the potential failure of the identity-of-indiscernibles. The topology induced by a quasi-metric is generally not symmetric, and often one considers left and right balls to capture directional structure.

Examples and constructions include distances in directed graphs, where the length of the shortest directed path from x to y yields a natural quasi-metric, and certain geometric metrics on convex domains (such as the Funk metric) that are inherently non-symmetric. A common way to recover a symmetric metric from a quasi-metric is to take the maximum or sum of d(x, y) and d(y, x).

Quasimetrics are studied in areas such as analysis, geometry, and theoretical computer science, where directional costs and asymmetry play a natural role. They form part of the broader framework of Lawvere metric spaces, which treats distance as a directed, order-theoretic notion.