A quasimetric space $(X,d)$ is a set $X$ together with a non-negative real-valued function $d: X \times X \longrightarrow \mathbb{R}$ (called a quasimetric) such that, for every $x,y,z \in X$ ,

• $d(x,y)\geq 0$ with equality if and only if $x=y$ .
• $d(x,z) \leq d(x,y) + d(y,z)$

In other words, a quasimetric space is a generalization of a metric space in which we drop the requirement that, for two points $x$ and $y$ , the ``distance'' between $x$ and $y$ is the same as the ``distance'' between $y$ and $x$ (i.e. the symmetry axiom of metric spaces).

Some properties:

• If $(X,d)$ is a quasimetric space, we can form a metric space $(X,d')$ where $d'$ is defined for all $x,y\in X$ by
  
  
• Every metric space is trivially a quasimetric space.
• A quasimetric that is symmetric (i.e. satisfies $d(x,y)=d(y,x)$ for all $x,y\in X$ is a metric.

Bibliography

1

L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.

2

Z. Shen, Lectures of Finsler geometry, World Sientific, 2001.