Space semimetric

In Mathematical, a space semimetric is a particular case of Espace prametric, which has only two of the three properties of a metric Espace, not requiring the checking of the triangular Inégalité.

Definition

A space semimetric $\backslash \; left\; (M,\; \backslash \; mathrm\; D\; \backslash \; right)$ is the data of a Ensemble $M$ is of a function $\backslash \; mathrm\; D:\; M\; \backslash \; times\; M\; \backslash \; to\; \backslash \; R^\; \{+\}$, called semimetric function (or semimetric ), which checks the following conditions:

• $\backslash \; mathrm\; D\; \backslash \; left\; (X,\; there\; \backslash \; right)\; \backslash \; Ge\; 0$ (positivity, inherited);
• $\backslash \; mathrm\; D\; \backslash \; left\; (X,\; there\; \backslash \; right)\; =\; 0\; \backslash \; mbox\; \{if\; and\; only\; if\}\; x=y$ (indiscernibility);
• $\backslash \; mathrm\; D\; \backslash \; left\; (X,\; there\; \backslash \; right)\; =\; \backslash \; mathrm\; D\; \backslash \; left\; (there,\; X\; \backslash \; right)$ (symmetry)

If the semimétrique one checks the triangular Inégalité, then it is a Métrique, and associated space becomes metric Espace.

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