Semimetric space

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.

Definition

A semimetric space (M,mathrm{d}) is a set M together with a function mathrm{d}:M imes M omathbb{R}^+ (called a semimetric) which satisfies the following conditions:
#,!mathrm{d}(x,y)ge0 ("non-negativity");
#,!mathrm{d}(x,y)=0mbox{ if and only if }x=y ("identity of indiscernibles");
#,!mathrm{d}(x,y)=mathrm{d}(y,x) ("symmetry")

References

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