Axiom:Metric Space Axioms/Also defined as

Metric Space Axioms: Also defined as

The numbering of the metric space axioms is arbitrary and varies between authors.

It is therefore a common practice, when referring to an individual axiom by number, to describe it briefly at the same time.

Some sources replace $(\text M 1)$ and $(\text M 4)$ with a combined axiom:

$(\text M 1')$

$:$

$\ds \map d {x, y} \ge 0; \quad \forall x, y \in A:$

$\ds \map d {x, y} = 0 \iff x = y $

thus allowing for there to be just three metric space axioms.

Others use:

$(\text M 1')$

$:$

$\ds \quad \forall x, y \in A:$

$\ds \map d {x, y} = 0 \iff x = y $

as the stipulation that $\map d {x, y} \ge 0$ can in fact be derived.