Espace quasi-métrique

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En mathématiques, la notion d'espace quasi-métrique^[1] généralise celle d'espace métrique. Les quasi-distances (ou quasi-métriques), non nécessairement symétriques, sont fréquentes dans la vie courante, mais rarement utilisées en mathématiques, et le vocabulaire est fluctuant^[2].

Définition

Une quasi-distance sur un ensemble $E$ est une fonction

\mathrm {d} :E\times E\to \mathbb {R} _{+}

telle que pour tout $x,y,z\in E$ ,

$\mathrm {d} (x,y)=0\iff x=y$ (séparation) ;
$\mathrm {d} (x,z)\leq \mathrm {d} (x,y)+\mathrm {d} (y,z)$ (inégalité triangulaire) ;

Un espace quasi-métrique est un couple $\left(E,\mathrm {d} \right)$ où $E$ est un ensemble et $\mathrm {d}$ est une quasi-distance sur $E$ .

Remarques :

Une quasi-distance symétrique est une distance.
Toute quasi-distance $\mathrm {d}$ induit une distance $\mathrm {d} '$ en posant :

\mathrm {d} '\left(x,y\right)={\frac {\mathrm {d} \left(x,y\right)+\mathrm {d} \left(y,x\right)}{2}}

Exemple

Considérons un système de routes, dont certaines sont éventuellement à sens unique. Le temps nécessaire pour aller d'un endroit à un autre en passant par la route donne une quasi-distance.

Notes et références

↑ J-M Huriot et J Perreur, « Distances, espaces et représentations (une revue) », 2017
↑ Les quasi-distances sont définies dans (en) Lynn Arthur Steen et J. Arthur Seebach, Jr., Counterexamples in Topology, New York, Dover, 1995, 2^e éd. (1^re éd. 1970), 244 p., poche (ISBN 978-0-486-68735-3, lire en ligne). Dans (en) Stefan Rolewicz, Functional Analysis and Control Theory : Linear Systems, Dordrecht, Springer, 1987, 524 p. (ISBN 978-90-277-2186-0, lire en ligne) elles sont appelées semi-distances, mais ce terme est déjà souvent utilisé pour deux autres généralisations de la notion d'espace métrique : voir espace semi-métrique et espace pseudo-métrique.

Liens externes

(en) « Quasimetric space », sur PlanetMath

(en) Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé « Quasimetric space » (voir la liste des auteurs).

Voir aussi

Portail des mathématiques

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