# hemimetric

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

1. 1.

$d(x,y)\geq 0$;

2. 2.

$d(x,z)\leq d(x,y)+d(y,z)$;

3. 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 $B(x,r)=\{y\in X:d(x,y) is the $r$-ball centered at $x$.

Title hemimetric Hemimetric 2013-03-22 14:24:12 2013-03-22 14:24:12 Koro (127) Koro (127) 5 Koro (127) Definition msc 54E25