pseudometric space

A pseudometric space is a set $X$ together with a non-negative real-valued function $d:X\times X\longrightarrow\mathbb{R}$ (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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$d(x,z)\leq d(x,y)+d(y,z)$

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	2013-03-22 14:40:18
Last modified on	2013-03-22 14:40:18
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	7
Author	mathcam (2727)
Entry type	Definition
Classification	msc 54E35
Synonym	pesudo-metric space
Related topic	MetricSpace
Related topic	QuasimetricSpace
Related topic	NormedVectorSpace
Related topic	Seminorm
Defines	pseudometric
Defines	pseudo-metric