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Revision difference : quasimetric space |
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Version 4 |
| 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 $x,y,z \in X$, |
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} |
\begin{itemize} |
| \item $d(x,y)\geq 0$ with equality if and only if $x=y$. |
\item $d(x,y)\geq 0$ with equality if and only if $x=y$. |
| \item $d(x,z) \leq d(x,y) + d(y,z)$ |
\item $d(x,z) \leq d(x,y) + d(y,z)$ |
| \end{itemize} |
\end{itemize} |
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| 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). |
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). |
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| Some properties: |
Some properties: |
| \begin{itemize} |
\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 |
\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*} |
\begin{align*} |
| d'(x,y) = \frac{1}{2}(d(x,y)+d(y,x)). |
d'(x,y) = \frac{1}{2}(d(x,y)+d(y,x)). |
| \end{align*} |
\end{align*} |
| \item Every metric space is trivially a quasimetric space. |
\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. |
\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} |
\end{itemize} |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{steen} L.A. Steen, J.A.Seebach, Jr., |
\bibitem{steen} L.A. Steen, J.A.Seebach, Jr., |
| \emph{Counterexamples in topology}, |
\emph{Counterexamples in topology}, |
| Holt, Rinehart and Winston, Inc., 1970. |
Holt, Rinehart and Winston, Inc., 1970. |
| \bibitem{shen} |
\bibitem{shen} |
| Z. Shen, \emph{Lectures of Finsler geometry}, World Sientific, 2001. |
Z. Shen, \emph{Lectures of Finsler geometry}, World Sientific, 2001. |
| \end{thebibliography} |
\end{thebibliography} |
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