Metric spaces

http://mathcs.chapman.edu/structuresold/files/Metric_spaces.pdf

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\begin{document}
\textbf{\Large Metric spaces}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Metric spaces}{edit}

\abbreviation{MetSp}

\begin{definition}
A \emph{metric space} is a structure $\mathbf{X}=\langle X,d\rangle$, where $d:X\times X\to [0,infty)$ is a \emph{distance metric}, i.e.,

points zero distance apart are identical: $d(x,y)=0\iff x=y$

$d$ is \emph{symmetric}:  $d(x,y)=d(y,x)$

the \emph{triangle inequality} holds: $d(x,z)\le d(x,y)+d(y,z)$

Remark: This is a template.
If you know something about this class, click on the 'Edit text of this page' link at the bottom and fill out this page.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
\end{definition}

\begin{morphisms}
Let $\mathbf{X}$ and $\mathbf{Y}$ be metric spaces. A morphism from $\mathbf{X}$ to $\mathbf{Y}$ is a function $h:X\rightarrow Y$ that is continuous in
the topology induced by the metric: $\forall z\in X\ \forall\epsilon>0\ \exists\delta>0\ \forall x\in X(0<d(x,z)<\delta\implies d(h(x),h(z))<\epsilon$
\end{morphisms}

\begin{definition}
An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle
...\rangle$ such that

$...$ is ...:  $axiom$
  
$...$ is ...:  $axiom$
\end{definition}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
\end{example}
\end{examples}

\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

\begin{tabular}{|ll|}\hline
  Classtype                       & higher-order \\\hline
  Amalgamation property           & \\\hline
  Strong amalgamation property    & \\\hline
  Epimorphisms are surjective     & \\\hline
\end{tabular}
\end{properties}
\end{table}

\begin{subclasses}\ 

  \href{Compact_metric_spaces.pdf}{Compact metric spaces}

\end{subclasses}

\begin{superclasses}\ 

  \href{Hausdorff_spaces.pdf}{Hausdorff spaces} reduced type

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Lastname19xx}
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 \href{http://www.ams.org/mathscinet-getitem?mr=12a:08034}{MRreview} 

\end{thebibliography}

\end{document}
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