# Difference between revisions of "Quasi-metric"

From Encyclopedia of Mathematics

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− | Let $\mathbb X$ | + | Let $\mathbb X$ be a nonempty set. A function $d:\mathbb{X}\times\mathbb{X}\to[0,\infty)$ which satisfies the following conditions for all $x,y\in\mathbb X$ |

1) $d(x,y)=0$ if and only if $x = y$ (the identity axiom); | 1) $d(x,y)=0$ if and only if $x = y$ (the identity axiom); | ||

− | 2) $d(x,y) + | + | 2) $d(x,y) + d(y,z) \geq d(x,z)$ (the triangle axiom); |

− | is called quasi-metric. A pair $(\mathbb X, d)$ is quasi-metric space. | + | is called a quasi-metric. A pair $(\mathbb X, d)$ is a quasi-metric space. |

− | The difference between [[Metric | 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 $x,y\in \mathbb X$ ). | + | The difference between a [[Metric | metric]] and a quasi-metric is that a 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=== | ||

+ | {| | ||

+ | |- | ||

+ | |valign="top"|{{Ref|Sch}}|| V. Schroeder, "Quasi-metric and metric spaces". Conform. Geom. Dyn. 10, 355 - 360 (2006) {{ZBL|1113.54014}} | ||

+ | |- | ||

+ | |valign="top"|{{Ref|Wil}}|| W. A. Wilson, "On Quasi-Metric Spaces". American Journal of Mathematics Vol. 53, No. 3 (1931), pp. 675-684 {{ZBL|0002.05503}} | ||

+ | |- | ||

+ | |} |

## Latest revision as of 19:38, 6 March 2016

Let $\mathbb X$ be a nonempty set. A function $d:\mathbb{X}\times\mathbb{X}\to[0,\infty)$ which satisfies the following conditions for all $x,y\in\mathbb X$

1) $d(x,y)=0$ if and only if $x = y$ (the identity axiom);

2) $d(x,y) + d(y,z) \geq d(x,z)$ (the triangle axiom);

is called a quasi-metric. A pair $(\mathbb X, d)$ is a quasi-metric space.

The difference between a metric and a quasi-metric is that a 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=29108