premmetric

Premmetric is a term used in topology and analysis to describe a distance-like function that generalizes the notion of a metric by dropping some of its axioms. Formally, a premmetric on a set X is a function d: X × X → [0, ∞) such that d(x, x) = 0 for all x in X. Unlike a metric, a premmetric is not required to be symmetric (d(x, y) may differ from d(y, x)) and it is not required to satisfy the triangle inequality. Consequently, many standard metric properties may fail.

A premmetric induces a natural topology on X. For each x ∈ X and r > 0, the open ball B(x, r) = { y ∈ X | d(x, y) < r } forms a basis for a topology, called the premetric topology. In general this topology can be non-Hausdorff or exhibit other unusual separation properties, depending on how d behaves. The same underlying set X equipped with a metric that satisfies the usual axioms is a special case of a premmetric space, where the stronger conditions hold.

Variants and relations to other notions: If a premmetric also satisfies symmetry and the triangle inequality, it becomes a metric. If it satisfies symmetry, the triangle inequality, and d(x, y) = 0 implies x = y, it is a metric in the standard sense; if symmetry is present but distance-zero points for distinct x and y are allowed, it is a pseudometric. When triangle inequality holds but symmetry is not required, the structure is related to a quasi-metric.

Applications of premetrics appear in areas where a distance-like tool is useful but full metric axioms are too strong, such as certain constructive frameworks, non-symmetric distance models, and discussions of topology induced by distance-like functions. See also: metric, pseudometric, quasi-metric, topology.