quasimetric space

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

• $d(x,y)\ge 0$ with equality if and only if $x=y$.

• $d(x,z)\le d(x,y)+d(y,z)$

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 $(X,d^{\prime })$ where $d^{\prime }$ is defined for all $x,y\in X$ by

  $d^{\prime }(x,y)={\displaystyle \frac{1}{2}}(d(x,y)+d(y,x)).$

• Every metric space is trivially a quasimetric space.

• A quasimetric that is (i.e. $d(x,y)=d(y,x)$ for all $x,y\in X$ is a metric.