hemimetric

A hemimetric on a set $X$ is a function $d:X\times X\to ℝ$ 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	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