Hemimetric space

In mathematics, a hemimetric space is a generalization of a metric space, obtained by removing the requirements of identity of indiscernibles and of symmetry. It is thus a generalization of both a quasimetric space and a pseudometric space, while being a special case of a prametric space.

Definition

A hemimetric on a set $X$ is a function $dcolon\; X\; imes\; X\; o\; mathbb\{R\}$ such that
#$,!\; d(x,y)geq\; 0$ (positivity);
#$,!\; d(x,z)\; leq\; d(x,y)\; +\; d(y,z)$ (subadditivity/triangle inequality);
#$,!\; d(x,x)=0$;for all $x,y,zin\; X$.

Hence, essentially $d$ is a metric which fails to satisfy symmetry and the property that distinct points have positive distance (the identity of indiscernibles).

A symmetric hemimetric is a pseudometric.

A hemimetric that can discern points is a quasimetric.

A hemimetric induces a topology on $X$ in the same way that a metric does, a basis of open sets being :$\{B\_r(x):\; xin\; X,\; r>0\},$

where

References

*planetmath|id=5903|title=hemimetric