Quasidistances

Quasidistances are distance-like constructs used to generalize the notion of distance by relaxing one or more axioms of a metric. On a set X, a quasidistance is a function d: X×X → [0, ∞) that typically satisfies nonnegativity (d(x,y) ≥ 0), the identity of indiscernibles in a weak form (often d(x,x) = 0 for all x), and the triangle inequality d(x,z) ≤ d(x,y) + d(y,z) for all x, y, z in X. A key departure from metric spaces is that symmetry is not required; in general, d(x,y) need not equal d(y,x). Some authors also allow d(x,y) = 0 with x ≠ y, leading to further related notions such as quasi-pseudometrics.

In many treatments, the term quasidistance overlaps with quasi-metrics, while a quasi-pseudometric drops the symmetry condition but keeps the triangle inequality and nonnegativity. Because of asymmetry, the topology induced by a quasidistance can be non-symmetric: forward and backward neighborhoods of a point may differ, and notions such as convergence and Cauchy sequences may have direction-dependent variants. Completeness and compactness concepts adapt to this asymmetric setting, yielding distinct analytical tools from the symmetric case.

Examples frequently cited include distances in directed graphs: the length of the shortest directed path from a to b defines a quasidistance on the graph’s vertices, satisfying triangle inequality but not necessarily symmetry. Quasidistances also arise in time-based or cost-based models where direction matters, such as propagation times or production costs that differ by direction.

Relation to metrics is by symmetrization: one may form a symmetric metric from a quasidistance with operations like max{d(x,y), d(y,x)} or d(x,y) + d(y,x). Quasidistances thus provide a flexible framework for modeling inherently directional or asymmetric phenomena.