Hemimetric space

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In mathematics, a hemimetric space is a generalization of a metric space, obtained by removing the requirement of discernability 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.

[edit] Definition

A hemimetric on a set $X$ is a function $d\colon X\times X\to \mathbb{R}$ 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 $x,y,z\in X$ .

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

$\{B_r(x): x\in X, r>0\},$