In mathematics, a hemimetric space is a generalization of a metric space, obtained by removing the requirement of discernability and of symmetry. It is thus a generalization of both a quasimetric space and a pseudometric space, while being a special case of a prametric space. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, a quasimetric space (M,d ) is a set M together with a function d : M × M -> R (called a quasimetric) which satisfies the following conditions: d(x, y) ≥ 0     (non-negativity) d(x, y) = 0   if and only if   x = y     (identity of indiscernibles) d(x, z... In mathematics, a metric space is a set (or space) where a distance between points is defined. ...

Definition

A hemimetric on a set X is a function such that

1. (positivity);
2. (subadditivity/triangle inequality);
3. ;

for all . A function f(x) is subadditive if for all x and y in the domain of f. ... In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ...

Hence, essentially d is a metric which fails to satisfy symmetry and the property that distinct points have positive distance (the identity of indiscernibles). The identity of indiscernibles is an ontological principle that states that if there is no way of telling two entities apart then they are one and the same entity. ...

A symmetric hemimetric is a pseudometric. In mathematics, a metric space is a set (or space) where a distance between points is defined. ...

A hemimetric that can discern points is a quasimetric.

A hemimetric induces a topology on X in the same way that a metric does, a basis of open sets being Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

where is the r-ball centered at x. A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ...

References

• This article incorporates material from hemimetric on PlanetMath, which is licensed under the GFDL.