generalization of a pseudometric

Let X be a set. Let d:X×X be a function with the property that d(x,y)0 for all x,yX. Then d is a

  1. 1.

    semi-pseudometric if d(x,y)=d(y,x) for all x,yX,

  2. 2.

    quasi-pseudometric if d(x,z)d(x,y)+d(y,z) for all x,y,zX.

X equipped with a function d described above is called a semi-pseudometric space or a quasi-pseudometric space, depending on whether d is a semi-pseudometric or a quasi-pseudometric. A pseudometric is the same as a semi-pseudometric that is a quasi-pseudometric at the same time.

If d satisfies the property that d(x,y)=0 implies x=y, then d is called a semi-metric if d is a semi-pseudometric, or a quasi-metric if d is a quasi-pseudometric.

Title generalizationPlanetmathPlanetmath of a pseudometric
Canonical name GeneralizationOfAPseudometric
Date of creation 2013-03-22 16:43:06
Last modified on 2013-03-22 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 semi-pseudometric space
Defines quasi-pseudometric space
Defines semi-pseudometric
Defines quasi-pseudometric