quasimetric space
A quasimetric space $(X,d)$ is a set $X$ together with a nonnegative realvalued function $d:X\times X\u27f6\mathbb{R}$ (called a quasimetric) such that, for every $x,y,z\in X$,

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

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$d(x,z)\le d(x,y)+d(y,z)$
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:

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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
${d}^{\prime}(x,y)={\displaystyle \frac{1}{2}}(d(x,y)+d(y,x)).$ 
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Every metric space is trivially a quasimetric space.

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A quasimetric that is (i.e. $d(x,y)=d(y,x)$ 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.
Title  quasimetric space 
Canonical name  QuasimetricSpace 
Date of creation  20130322 14:40:21 
Last modified on  20130322 14:40:21 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  8 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 54E35 
Synonym  quasimetric space 
Related topic  PseudometricSpace 
Related topic  MetricSpace 
Related topic  GeneralizationOfAPseudometric 
Defines  quasimetric 
Defines  quasimetric 