Quasi-metric
From Encyclopedia of Mathematics
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
Let $\mathbb X$ is a nonempty set. A function $d:\mathbb{X}\times\mathbb{X}\to[0,\infty)$ which satisfies following conditions for all $x,y\in\mathbb X$
1) $d(x,y)=0$ if and only if $x = y$ (the identity axiom);
2) $d(x,y) + \rho(y,z) \geq d(x,z)$ (the triangle axiom);
is called quasi-metric. A pair $(\mathbb X, d)$ is quasi-metric space.
The difference between metric and quasi-metric is that quasi-metric does not possess the symmetry axiom (in the case we allow $d(x,y)\ne d(y,x)$ for some $x,y\in \mathbb X$ ).
Reference
[Sch] | V. Schroeder, "Quasi-metric and metric spaces". Conform. Geom. Dyn. 10, 355 - 360 (2006) Zbl 1113.54014 |
[Wil] | W. A. Wilson, "On Quasi-Metric Spaces". American Journal of Mathematics Vol. 53, No. 3 (1931), pp. 675-684 Zbl 0002.05503 |
How to Cite This Entry:
Quasi-metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-metric&oldid=29111
Quasi-metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-metric&oldid=29111