Semimetric space

Semimetric space
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Semimetric_space".

In topology, a semimetric space is a generalized metric space in which the triangle inequality is not required. In translations of Russian texts, a semimetric is sometimes called a symmetric.

Note: In functional analysis and related mathematical disciplines, the word semimetric space is used as a synonym for pseudometric space, because every seminorm induces a pseudometric.

The ultrametric and inframetric replace the triangle inequality with a stronger form.

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)

References

Arkhangel'skii, A.V.; Pontryagin, L.S. (1990). General Topology I: Basic Concepts and Constructions Dimension Theory, Encyclopaedia of Mathematical Sciences. Springer. ISBN 3-540-18178-4.
Steen, Lynn Arthur; Seebach, Arthur [1970] (1995). Counterexamples in Topology. Dover Publications. ISBN 048668735X.

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