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 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 $(M1)$ and $(M4)$ with a combined axiom:
 * $(M1'): \quad d \left({x, y}\right) \ge 0; \quad \forall x, y \in X: d \left({x, y}\right) = 0 \iff x = y$

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

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

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

Also see

 * Distance Function for Distinct Elements in Metric Space is Strictly Positive