Mathematical Structures: Metric spaces

# Metric spaces

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Changed: 92c92
 \href{Hausdorff_spaces.pdf}{Hausdorff_spaces} reduced type
 \href{Hausdorff_spaces.pdf}{Hausdorff spaces} reduced type

http://mathcs.chapman.edu/structuresold/files/Metric_spaces.pdf
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\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
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\begin{document}
\textbf{\Large Metric spaces}

\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.

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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