pseudometric spacePseudometricSpace2004-10-02 13:33:432004-10-07 11:39:55Definitionpseudometricpseudo-metric% this is the default PlanetMath preamble. as your knowledge
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\renewcommand{\>}{\rangle}A {\em pseudometric space} is a set $X$ together with a non-negative real-valued function $d: X \times X \longrightarrow \mathbb{R}$ (called a {\em pseudometric}) such that, for every $x,y,z \in X$,
\begin{itemize}
\item $d(x,x) = 0$.
\item $d(x,y) = d(y,x)$
\item $d(x,z) \leq d(x,y) + d(y,z)$
\end{itemize}
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 $y$.
\begin{thebibliography}{9}
\bibitem{steen} L.A. Steen, J.A.Seebach, Jr.,
\emph{Counterexamples in topology},
Holt, Rinehart and Winston, Inc., 1970.
\end{thebibliography}