# Definition:Metric Space/Triangle Inequality

## Definition

Let $M = \struct {A, d}$ be a metric space, satisfying the metric space axioms:

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

Axiom $\text M 2$ is referred to as the triangle inequality, as it is a generalization of the Triangle Inequality which holds on the real number line and complex plane.

## Examples

### $4$ Points

Let $M = \struct {A, d}$ be a metric space.

Let $x, y, z, t \in A$.

Then:

$\map d {x, z} + \map d {y, t} \ge \size {\map d {x, y} - \map d {z, t} }$

## Also see

$\forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$