Quasimetric space

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In mathematics, a quasimetric space generalizes the idea of a metric space by removing the requirement of symmetry of the metric. A quasimetric space is a special case of a hemimetric space, to which the requirement of distinguishability is added.

[edit] Definition

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

  1. \,\!\mathrm{d}(x,y)\ge0 (non-negativity);
  2. \,\!\mathrm{d}(x,y)=0\mbox{ if and only if }x=y (identity of indiscernibles);
  3. \,\!\mathrm{d}(x,z)\le\mathrm{d}(x,y)+\mathrm{d}(y,z) (subadditivity/triangle inequality).

If (M,d) is a quasimetric space, a metric space (M,d') can be formed by taking

\mathrm{d}'(x,y)=\frac{(\mathrm{d}(x,y)+d(y,x))}{2}.

[edit] See also

[edit] References

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