Definition:Metric Space/Triangle Inequality

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Definition

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

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


Axiom $M2$ 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.


Also see

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


Sources