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 $(M,d)$ is a set $M$ together with a function $\mathrm{d}:M\times M\to\mathbb{R}$ (called a quasimetric) which satisfies the following conditions:

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