# 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 $x,y,z\in X$,

• $d(x,x)=0$.

• $d(x,y)=d(y,x)$

• $d(x,z)\leq d(x,y)+d(y,z)$

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