quasimetric space

A quasimetric space $(X,d)$ is a set $X$ together with a non-negative real-valued 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 (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 2013-03-22 14:40:21 Last modified on 2013-03-22 14:40:21 Owner mathcam (2727) Last modified by mathcam (2727) Numerical id 8 Author mathcam (2727) Entry type Definition Classification msc 54E35 Synonym quasi-metric space Related topic PseudometricSpace Related topic MetricSpace Related topic GeneralizationOfAPseudometric Defines quasimetric Defines quasi-metric