# Quasi-metric

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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 $x,y\in \mathbb X$ ).

How to Cite This Entry:
Quasi-metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-metric&oldid=29108