Axiom:Quasimetric Axioms

Definition

Let $A$ be a set.

Let $d: A \times A \to \R$ be a real-valued function.

$d$ is a quasimetric on the set $A$ if and only if $d$ satisfies the axioms:

\((\text M 1)\)	$:$			\(\ds \forall x \in A:\)	\(\ds \map d {x, x} = 0 \)
\((\text M 2)\)	$:$			\(\ds \forall x, y, z \in A:\)	\(\ds \map d {x, y} + \map d {y, z} \ge \map d {x, z} \)
\((\text M 4)\)	$:$			\(\ds \forall x, y \in A:\)	\(\ds x \ne y \implies \map d {x, y} > 0 \)

These criteria are called the quasimetric axioms.

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

Also see

• Definition:Quasimetric

• Definition:Quasimetric Space