Definition:Metric Space/Triangle Inequality

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

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

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.


$4$ Points

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

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


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

Also see

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