Hemimetric space

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In mathematics, a hemimetric space is a generalization of a metric space, obtained by removing the requirements of identity of indiscernibles 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 <math>X</math> is a function <math>d\colon X\times X\to \mathbb{R}</math> such that

<math>\,\! d(x,y)\geq 0</math> (positivity);
<math>\,\! d(x,z) \leq d(x,y) + d(y,z)</math> (subadditivity/triangle inequality);
<math>\,\! d(x,x)=0</math>;

for all <math>x,y,z\in X</math>.

Hence, essentially <math>d</math> 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 <math>X</math> in the same way that a metric does, a basis of open sets being

where <math>B_r(x)=\{y\in X : d(x,y)<r\}</math> is the r-ball centered at <math>x</math>.