hemimetric
A hemimetric on a set $X$ is a function $d:X\times X\to \mathbb{R}$ such that

1.
$d(x,y)\ge 0$;

2.
$d(x,z)\le d(x,y)+d(y,z)$;

3.
$d(x,x)=0$;
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. A hemimetric induces a topology on $X$ in the same way that a metric does, a basis of open sets being
$$\{B(x,r):x\in X,r>0\},$$ 
where $$ is the $r$ball centered at $x$.
Title  hemimetric 

Canonical name  Hemimetric 
Date of creation  20130322 14:24:12 
Last modified on  20130322 14:24:12 
Owner  Koro (127) 
Last modified by  Koro (127) 
Numerical id  5 
Author  Koro (127) 
Entry type  Definition 
Classification  msc 54E25 