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