# Category:Definitions/Metric Spaces

A metric space $M = \struct {A, d}$ is an ordered pair consisting of:
$(1): \quad$ a non-empty set $A$
$(2): \quad$ a real-valued function $d: A \times A \to \R$ which acts on $A$, satisfying the metric space axioms:
 $(\text M 1)$ $:$ $\ds \forall x \in A:$ $\ds \map d {x, x} = 0$ $(\text M 2)$ $:$ $\ds \forall x, y, z \in A:$ $\ds \map d {x, y} + \map d {y, z} \ge \map d {x, z}$ $(\text M 3)$ $:$ $\ds \forall x, y \in A:$ $\ds \map d {x, y} = \map d {y, x}$ $(\text M 4)$ $:$ $\ds \forall x, y \in A:$ $\ds x \ne y \implies \map d {x, y} > 0$