generalization of a pseudometric
Let $X$ be a set. Let $d:X\times X\to \mathbb{R}$ be a function with the property that $d(x,y)\ge 0$ for all $x,y\in X$. Then $d$ is a

1.
semipseudometric if $d(x,y)=d(y,x)$ for all $x,y\in X$,

2.
quasipseudometric if $d(x,z)\le d(x,y)+d(y,z)$ for all $x,y,z\in X$.
$X$ equipped with a function $d$ described above is called a semipseudometric space or a quasipseudometric space, depending on whether $d$ is a semipseudometric or a quasipseudometric. A pseudometric is the same as a semipseudometric that is a quasipseudometric at the same time.
If $d$ satisfies the property that $d(x,y)=0$ implies $x=y$, then $d$ is called a semimetric if $d$ is a semipseudometric, or a quasimetric if $d$ is a quasipseudometric.
Title  generalization^{} of a pseudometric 
Canonical name  GeneralizationOfAPseudometric 
Date of creation  20130322 16:43:06 
Last modified on  20130322 16:43:06 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  6 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 54E35 
Synonym  semipseudometric 
Synonym  quasipseudometric 
Synonym  semipseudometric space 
Synonym  quasipseudometric space 
Related topic  semimetric 
Related topic  quasimetric 
Related topic  GeneralizationOfAUniformity 
Defines  semipseudometric space 
Defines  quasipseudometric space 
Defines  semipseudometric 
Defines  quasipseudometric 