Hemimetric space
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In mathematics, a hemimetric space is a generalization of a metric space, obtained by removing the requirements 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.
[edit] Definition
A hemimetric on a set X is a function 
such that
(positivity);
(subadditivity/triangle inequality);
;
for all 
.
Hence, essentially d is a metric which fails to satisfy symmetry and the property that distinct points have positive distance (the identity of indiscernibles).
A symmetric hemimetric is a pseudometric.
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
where 
is the r-ball centered at x.
[edit] References
- This article incorporates material from hemimetric on PlanetMath, which is licensed under the GFDL.
