Quasimetric space

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In mathematics, a quasimetric space generalizes the idea of a metric space by removing the requirement of symmetry of the metric. A quasimetric space is a special case of a hemimetric space, to which the requirement of distinguishability is added.

[edit] Definition

A quasimetric space <math>(M,\mathrm{d})</math> is a set <math>M</math> together with a function <math>\mathrm{d}:M\times M\to\mathbb{R}</math> (called a quasimetric) which satisfies the following conditions:

<math>\,\!\mathrm{d}(x,y)\ge0</math> (non-negativity);
<math>\,\!\mathrm{d}(x,y)=0\mbox{ if and only if }x=y</math> (identity of indiscernibles);
<math>\,\!\mathrm{d}(x,z)\le\mathrm{d}(x,y)+\mathrm{d}(y,z)</math> (subadditivity/triangle inequality).

If <math>(M,\mathrm{d})</math> is a quasimetric space, a metric space <math>(M,\mathrm{d}')</math> can be formed by taking

<math>\mathrm{d}'(x,y)=\frac{(\mathrm{d}(x,y)+d(y,x))}{2}</math>.