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quasimetric space
A quasimetric space $(X,d)$ is a set $X$ together with a nonnegative realvalued function $d:X\times X\longrightarrow\mathbb{R}$ (called a quasimetric) such that, for every $x,y,z\in X$,

$d(x,y)\geq 0$ with equality if and only if $x=y$.

$d(x,z)\leq d(x,y)+d(y,z)$
In other words, a quasimetric space is a generalization of a metric space in which we drop the requirement that, for two points $x$ and $y$, the “distance” between $x$ and $y$ is the same as the “distance” between $y$ and $x$ (i.e. the symmetry axiom of metric spaces).
Some properties:

If $(X,d)$ is a quasimetric space, we can form a metric space $(X,d^{{\prime}})$ where $d^{{\prime}}$ is defined for all $x,y\in X$ by
$\displaystyle d^{{\prime}}(x,y)=\frac{1}{2}(d(x,y)+d(y,x)).$ 
Every metric space is trivially a quasimetric space.

A quasimetric that is symmetric (i.e. satisfies $d(x,y)=d(y,x)$ for all $x,y\in X$ is a metric.
References
 1 L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.
 2 Z. Shen, Lectures of Finsler geometry, World Sientific, 2001.
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