Quasimetric

A quasimetric on a set X is a function d: X × X → [0, ∞) that satisfies nonnegativity, identity on the diagonal, and the triangle inequality, but does not require symmetry. Concretely, for all x, y, z in X, one has d(x, x) = 0, d(x, y) ≥ 0, and d(x, z) ≤ d(x, y) + d(y, z). Unlike a metric, it is not required that d(x, y) = d(y, x); in general the distance from x to y can differ from the distance from y to x. Some authors also require the identity of indiscernibles in the sense that d(x, y) = 0 implies x = y, but this is not universal in all definitions of a quasimetric.

Quasimetric spaces induce asymmetric topologies; the open balls B_d(x, r) = {y ∈ X : d(x, y) < r} need not be symmetric with respect to exchanging x and y. This asymmetry is useful in contexts where direction or order matters, such as time, precedence, or directed costs.

Typical examples include the distance in a directed graph with nonnegative edge lengths, where d(u, v) is the length of the shortest directed path from u to v; this distance may differ from d(v, u) and may be infinite if no directed path exists. Quasimetric notions also arise in domain theory and related areas of theoretical computer science.

Relation to metric spaces: every metric is a quasimetric, since it satisfies the same axioms plus symmetry. A common way to obtain a symmetric metric from a quasimetric is to define d′(x, y) = d(x, y) + d(y, x); this function is symmetric and satisfies the triangle inequality, hence is a metric. Another common symmetricization is d_s(x, y) = max{d(x, y), d(y, x)} under suitable conditions.