quasimetric spaceQuasimetricSpace2004-10-02 13:40:512006-06-17 00:12:06Definitionquasimetricquasi-metric% this is the default PlanetMath preamble. as your knowledge
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\renewcommand{\>}{\rangle}A {\em quasimetric space} $(X,d)$ is a set $X$ together with a non-negative real-valued function $d: X \times X \longrightarrow \mathbb{R}$ (called a {\em quasimetric}) such that, for every $x,y,z \in X$,
\begin{itemize}
\item $d(x,y)\geq 0$ with equality if and only if $x=y$.
\item $d(x,z) \leq d(x,y) + d(y,z)$
\end{itemize}
In other words, a quasimetric space is a generalization of a metric space in which we drop the requirement that, for two points $x$ and $y$, the ``distance'' between $x$ and $y$ is the same as the ``distance'' between $y$ and $x$ (i.e. the symmetry axiom of metric spaces).
Some properties:
\begin{itemize}
\item If $(X,d)$ is a quasimetric space, we can form a metric space $(X,d')$ where $d'$ is defined for all $x,y\in X$ by
\begin{align*}
d'(x,y) = \frac{1}{2}(d(x,y)+d(y,x)).
\end{align*}
\item Every metric space is trivially a quasimetric space.
\item A quasimetric that is \PMlinkescapetext{symmetric} (i.e. \PMlinkescapetext{satisfies} $d(x,y)=d(y,x)$ for all $x,y\in X$ is a metric.
\end{itemize}
\begin{thebibliography}{9}
\bibitem{steen} L.A. Steen, J.A.Seebach, Jr.,
\emph{Counterexamples in topology},
Holt, Rinehart and Winston, Inc., 1970.
\bibitem{shen}
Z. Shen, \emph{Lectures of Finsler geometry}, World Sientific, 2001.
\end{thebibliography}