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\import{topology/topological-space.tex}
\import{numbers.tex}
\import{function.tex}
\import{set/powerset.tex}
\import{topology/basis.tex}
\section{Metric Spaces}
\begin{definition}\label{metric}
$f$ is a metric on $M$ iff $f$ is a function from $M \times M$ to $\reaaals$ and
for all $x,y,z \in M$ we have
$f(x,x) = \zero$ and
$f(x,y) = f(y,x)$ and
$f(x,y) \leq f(x,z) + f(z,y)$ and
if $x \neq y$ then $\zero < f(x,y)$.
\end{definition}
\begin{definition}\label{open_ball}
$\openball{r}{x}{f} = \{z \in M \mid \text{ $f$ is a metric on $M$ and $f(x,z) < r$ } \}$.
\end{definition}
\begin{definition}\label{set_of_balls}
$\balls{d}{M} = \{ O \in \pow{M} \mid \text{there exists $x,r$ such that $r \in \reals$ and $x \in M$ and $O = \openball{r}{x}{d}$ } \}$.
\end{definition}
\begin{definition}\label{metricopens}
$\metricopens{d}{M} = \genOpens{\balls{d}{M}}{M}$.
\end{definition}
%TODO: \metric_opens{d} = {hier die construction für topology} DONE.
%TODO: Die induzierte topology definieren und dann in struct verwenden.
\begin{struct}\label{metric_space}
A metric space $M$ is a onesorted structure equipped with
\begin{enumerate}
\item $\metric$
\end{enumerate}
such that
\begin{enumerate}
\item \label{metric_space_metric} $\metric[M]$ is a metric on $M$.
\item \label{metric_space_topology} $M$ is a topological space.
\item \label{metric_space_opens} $\metricopens{ \metric[M] }{M} = \opens[M]$.
\end{enumerate}
\end{struct}
\begin{abbreviation}\label{descriptive_syntax_for_openball1}
$U$ is an open ball in $M$ of $x$ with radius $r$ iff $x \in M$ and $M$ is a metric space and $U = \openball{r}{x}{\metric[M]}$.
\end{abbreviation}
\begin{abbreviation}\label{descriptive_syntax_for_openball2}
$U$ is an open ball in $M$ iff there exist $x \in M$ such that there exist $r \in \reals$ such that $U$ is an open ball in $M$ of $x$ with radius $r$.
\end{abbreviation}
\begin{lemma}\label{union_of_open_balls_is_open}
Let $M$ be a metric space.
For all $U,V \subseteq M$ if $U$, $V$ are open balls in $M$ then $U \union V$ is open in $M$.
\end{lemma}
\begin{lemma}\label{metric_implies_topology}
Let $M$ be a set, and let $f$ be a metric on $M$.
Then $M$ is a metric space.
\end{lemma}
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