Quasimetric space

In mathematics, a quasimetric space is a generalized metric space in which the metric is not necessarily symmetric. Although quasimetrics are common in real life, this notion is rarely used in mathematics, and its name is not entirely standardized. [Quasimetrics are defined in Steen and Seebach (1995). In Rolewicz (1987) they are called "semimetric". This is unfortunate since this term is already in frequent use for two other generalizations of metrics.]

Definition

A quasimetric space $(M,mathrm{d})$ is a set $M$ together with a function $mathrm{d}:M imes M omathbb{R}$ (called a quasimetric) which satisfies the following conditions:
# $,!mathrm{d}(x,y)ge0$ ("non-negativity");
# $,!mathrm{d}(x,y)=0mbox{ if and only if }x=y$ ("identity of indiscernibles");
# $,!mathrm{d}(x,z)lemathrm{d}(x,y)+mathrm{d}(y,z)$ ("subadditivity"/"triangle inequality").

If $(M,mathrm{d})$ is a quasimetric space, a metric space $(M,mathrm{d}')$ can be formed by taking: $mathrm{d}'(x,y)=frac{(mathrm{d}(x,y)+mathrm{d}(y,x))}{2}$ .

Example

A set of mountain villages with "d(x,y)" being the average time it takes to walk from village "x" to village "y".

Notes

References

Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

Hemimetric space — In mathematics, a hemimetric space is a generalization of a metric space, obtained by removing the requirements of identity of indiscernibles and of symmetry. It is thus a generalization of both a quasimetric space and a pseudometric space, while … Wikipedia
Metric space — In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined. The metric space which most closely corresponds to our intuitive understanding of space is the 3 dimensional Euclidean… … Wikipedia
Sierpiński space — In mathematics, Sierpiński space (or the connected two point set) is a finite topological space with two points, only one of which is closed.It is the smallest example of a topological space which is neither trivial nor discrete. It is named… … Wikipedia
Prametric space — In topology, a prametric space generalizes the concept of a metric space by not requiring the conditions of symmetry, indiscernability and the triangle inequality. Prametric spaces occur naturally as maps between metric spaces.DefinitionA… … Wikipedia
Metric (mathematics) — In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric … Wikipedia
Espace Quasimétrique — En mathématiques, un espace quasimétrique est un cas particulier d espace hémimétrique, qui généralise ainsi la notion d espace métrique ne nécessitant pas la condition de symétrie. De tels espaces apparaissent parfois en théorie des graphes.… … Wikipédia en Français
Espace quasimetrique — Espace quasimétrique En mathématiques, un espace quasimétrique est un cas particulier d espace hémimétrique, qui généralise ainsi la notion d espace métrique ne nécessitant pas la condition de symétrie. De tels espaces apparaissent parfois en… … Wikipédia en Français
Espace quasimétrique — En mathématiques, la notion d espace quasimétrique généralise celle d espace métrique. Les quasidistances ou quasimétriques, non nécessairement symétriques, sont fréquentes dans la vie courante, mais rarement utilisées en mathématiques, et le… … Wikipédia en Français
Quasimétrique — Espace quasimétrique En mathématiques, un espace quasimétrique est un cas particulier d espace hémimétrique, qui généralise ainsi la notion d espace métrique ne nécessitant pas la condition de symétrie. De tels espaces apparaissent parfois en… … Wikipédia en Français
List of mathematics articles (Q) — NOTOC Q Q analog Q analysis Q derivative Q difference polynomial Q exponential Q factor Q Pochhammer symbol Q Q plot Q statistic Q systems Q test Q theta function Q Vandermonde identity Q.E.D. QED project QR algorithm QR decomposition Quadratic… … Wikipedia

Academic Dictionaries and Encyclopedias

Quasimetric space

Look at other dictionaries:

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

Quasimetric space

Look at other dictionaries:

Share the article and excerpts

Direct link