Definition:Metric Space Axioms

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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:

\((M1)\)   $:$     \(\displaystyle \forall x \in A:\) \(\displaystyle \map d {x, x} = 0 \)             
\((M2)\)   $:$     \(\displaystyle \forall x, y, z \in A:\) \(\displaystyle \map d {x, y} + \map d {y, z} \ge \map d {x, z} \)             
\((M3)\)   $:$     \(\displaystyle \forall x, y \in A:\) \(\displaystyle \map d {x, y} = \map d {y, x} \)             
\((M4)\)   $:$     \(\displaystyle \forall x, y \in A:\) \(\displaystyle x \ne y \implies \map d {x, y} > 0 \)             


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 $(M1)$ and $(M4)$ with a combined axiom:

$(M1'): \quad \map d {x, y} \ge 0; \quad \forall x, y \in X: \map d {x, y} = 0 \iff x = y$

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


Others use:

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

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


Also see


Sources