hemimetric

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

1.

$d(x,y)\geq 0$ ;
2.

$d(x,z)\leq d(x,y)+d(y,z)$ ;
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 $B(x,r)=\{y\in X:d(x,y)<r\}$ is the $r$ -ball centered at $x$ .

Title	hemimetric
Canonical name	Hemimetric
Date of creation	2013-03-22 14:24:12
Last modified on	2013-03-22 14:24:12
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	5
Author	Koro (127)
Entry type	Definition
Classification	msc 54E25