Definition:Quasimetric

Definition

A quasimetric on a set $X$ is a real-valued function $d: X \times X \to \R$ which satisfies the following conditions for all $x, y, z \in X$:

M1: $d \left({x, x}\right) = 0$

M2: $d \left({x, y}\right) + d \left({y, z}\right) \ge d \left({x, z}\right)$

M4: $x \ne y \implies d \left({x, y}\right) > 0$

Note the numbering system of these conditions. They are numbered this way so as to retain consistency with the metric space axioms, of which these are a subset.

Quasimetric Space

A quasimetric space $M = \left({X, d}\right)$ is an ordered pair consisting of a set $X \ne \varnothing$ followed by a quasimetric $d: X \times X \to \R$ which acts on that set.

Also see

• Compare this definition with that for a metric.

The difference between a quasimetric and a metric is that a quasimetric does not insist that the distance function between distinct points is commutative, that is, that $d \left({x, y}\right) = d \left({y, x}\right)$.

• Pseudometric

• Results about quasimetric spaces can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 5$