semimetrics

Semimetric is a distance-like function defined on a set X. A semimetric d: X × X → [0, ∞) is typically required to satisfy: d(x, y) ≥ 0 for all x, y; d(x, x) = 0 for all x; and d(x, y) = d(y, x) for all x, y (symmetry). Unlike a metric, a semimetric is not generally required to satisfy the triangle inequality. Some authors also allow d(x, y) = 0 for distinct points. In standard terminology, a metric satisfies all the usual axioms including the triangle inequality and identity of indiscernibles (d(x, y) = 0 only if x = y). When the triangle inequality fails, the function is typically described as a semimetric; when the identity of indiscernibles is also relaxed, it may be called a pseudometric in some texts.

A common simple semimetric that is not a metric can be defined on a three-point set {a, b, c} by setting d(a, a) = d(b, b) = d(c, c) = 0, d(a, b) = 1, d(b, c) = 1, and d(a, c) = 3. This function is nonnegative, symmetric, and vanishes on the diagonal, but it violates the triangle inequality since d(a, c) > d(a, b) + d(b, c).

Semimetrics are used in contexts where a symmetric, nonnegative distance-like measure is useful but the triangle inequality is not guaranteed or required. They induce a topology in a manner similar to metrics, though some standard metric-space results may not hold. Related concepts include metrics, pseudometrics, and quasi-metrics.