Definition:Metric Space/Triangle Inequality

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Definition

Let $M = \left({A, d}\right)$ be a metric space, satisfying the metric space axioms:

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


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

$\forall x, y, z \in X: \left|{d \left({x, z}\right) - d \left({y, z}\right)}\right| \le d \left({x, y}\right)$


Sources