pseudometric space
A pseudometric space is a set $X$ together with a nonnegative realvalued function $d:X\times X\u27f6\mathbb{R}$ (called a pseudometric) such that, for every $x,y,z\in X$,

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$d(x,x)=0$.

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$d(x,y)=d(y,x)$

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$d(x,z)\le d(x,y)+d(y,z)$
In other words, a pseudometric space is a generalization^{} of a metric space in which we allow the possibility that $d(x,y)=0$ for distinct values of $x$ and $y$.
References
 1 L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.
Title  pseudometric space 
Canonical name  PseudometricSpace 
Date of creation  20130322 14:40:18 
Last modified on  20130322 14:40:18 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  7 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 54E35 
Synonym  pesudometric space 
Related topic  MetricSpace 
Related topic  QuasimetricSpace 
Related topic  NormedVectorSpace 
Related topic  Seminorm 
Defines  pseudometric 
Defines  pseudometric^{} 