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

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