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.

Definition

A quasimetric space $(M,\mathrm {d} )$ is a set $M$ together with a function $\mathrm {d} :M\times M\to \mathbb {R}$ (called a quasimetric) which satisfies the following conditions:

1. $\,\!\mathrm {d} (x,y)\geq 0$ (non-negativity);
2. $\,\!\mathrm {d} (x,y)=0{\mbox{ if and only if }}x=y$ (identity of indiscernibles);
3. $\,\!\mathrm {d} (x,z)\leq \mathrm {d} (x,y)+\mathrm {d} (y,z)$ (subadditivity/triangle inequality).

If $(M,\mathrm {d} )$ is a quasimetric space, a metric space $(M,\mathrm {d} ')$ can be formed by taking

$\mathrm {d} '(x,y)={\frac {(\mathrm {d} (x,y)+d(y,x))}{2}}$.

Example

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

References

• Rolewicz, Stefan (1987). Functional Analysis and Control Theory: Linear Systems. Springer. ISBN 9027721866.
• Steen, Lynn Arthur (1995) [1970]. Counterexamples in Topology. Dover Publications. ISBN 048668735X. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• "quasimetric space". PlanetMath.