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:


 * $(M1'): \quad \forall x, y \in X: d \left({x, y}\right) = 0 \iff x = y$

which leads to an equivalent system of axioms $(M1')$, $(M2)$, $(M3)$, which together imply in particular that $d$ is non-negative:

indeed, by symmetry and triangle inequality:


 * $d(x, y) + d(x, y) = d(x, y) + d(y, x) \ge d(x, x) = 0$

hence


 * $d(x, y) \ge 0$

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.