Axiom:Metric Space Axioms

Definition
Let $A$ be a set upon which a distance function $d: A \times A \to \R$ is imposed.

The metric space axioms are the conditions on $d$ which are satisfied for all elements of $A$ in order to make $\left({A, d}\right)$ a metric space:

Also defined as
The numbering of the axioms varies between authors.

Some merge M1 and M4. Some keep the same axioms but number them differently.

Some replace M1 and M4 with:

which can be seen trivially to be equivalent to M1 and M4.
 * M1': $\forall x, y \in X: d \left({x, y}\right) = 0 \iff x = y$
 * M4': $\forall x, y \in X: d \left({x, y}\right) \ge 0$

Some sources number the axioms $M0$ to $M3$.

The numbering is to a certain extent arbitrary.

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