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A quasimetric on a set $X$ is a real-valued function $d: A \times A \to \R$ which satisfies the following conditions:

\((M1)\)   $:$     \(\displaystyle \forall x \in A:\) \(\displaystyle d \left({x, x}\right) = 0 \)             
\((M2)\)   $:$     \(\displaystyle \forall x, y, z \in A:\) \(\displaystyle d \left({x, y}\right) + d \left({y, z}\right) \ge d \left({x, z}\right) \)             
\((M4)\)   $:$     \(\displaystyle \forall x, y \in A:\) \(\displaystyle x \ne y \implies d \left({x, y}\right) > 0 \)             

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.

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

Quasimetric Space

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

Also known as

A quasimetric on a quasimetric space can be referred to as a distance function in the same way as a metric on a metric space.

Also see

  • Results about quasimetric spaces can be found here.