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 for $\struct {A, d}$ to be a metric space:

 $(\text M 1)$ $:$ $\ds \forall x \in A:$ $\ds \map d {x, x} = 0$ $(\text M 2)$ $:$ Triangle Inequality: $\ds \forall x, y, z \in A:$ $\ds \map d {x, y} + \map d {y, z} \ge \map d {x, z}$ $(\text M 3)$ $:$ $\ds \forall x, y \in A:$ $\ds \map d {x, y} = \map d {y, x}$ $(\text M 4)$ $:$ $\ds \forall x, y \in A:$ $\ds x \ne y \implies \map d {x, y} > 0$

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 $(\text M 1)$ 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.