Semimetric space

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In mathematics, a semimetric space generalizes the concept of a metric space by not requiring the condition of satisfying the triangle inequality. Thus, a semimetric space is a special case of a prametric space, being defined by a symmetric, discernible prametric. Because of its symmetry properties, in translations of Russian texts, a semimetric is sometimes called a symmetric.

[edit] Definition

A semimetric space $(M,d)$ is a set $M$ together with a function $\mathrm{d}:M\times M\to\mathbb{R}^+$ (called a semimetric) which satisfies the following conditions:

$\,\!\mathrm{d}(x,y)\ge0$ (non-negativity);
$\,\!\mathrm{d}(x,y)=0\mbox{ if and only if }x=y$ (identity of indiscernibles);
$\,\!\mathrm{d}(x,y)=\mathrm{d}(y,x)$ (symmetry)