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\import{topology/topological-space.tex}
\import{numbers.tex}
\import{function.tex}
\section{Metric Spaces}
\begin{abbreviation}\label{metric}
$f$ is a metric iff $f$ is a function to $\reals$.
\end{abbreviation}
\begin{axiom}\label{metric_axioms}
$f$ is a metric iff $\dom{f} = A \times A$ and
for all $x,y,z \in A$ 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{axiom}
\begin{definition}\label{open_ball}
$\openball{r}{x}{f} = \{z \in M \mid \text{ $f$ is a metric and $\dom{f} = M \times M$ and $f(x,z)<r$ } \}$.
\end{definition}
\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_d} $\metric[M]$ is a function from $M \times M$ to $\reals$.
\item \label{metric_space_metric} $\metric[M]$ is a metric.
\item \label{metric_space_topology} $M$ is a topological space.
\item \label{metric_space_opens} for all $x \in M$ for all $r \in \reals$ $\openball{r}{x}{\metric[M]} \in \opens[M]$.
\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, let $U$ be an open ball in $M$, and let
$V$ be an open ball in $M$.
Then $U \union V$ is open in $M$.
\end{lemma}
%\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_d} $\metric[M]$ is a function from $M \times M$ to $\reals$.
% \item \label{metric_space_distence_of_a_point} $\metric[M](x,x) = \zero$.
% \item \label{metric_space_positiv} for all $x,y \in M$ if $x \neq y$ then $\zero < \metric[M](x,y)$.
% \item \label{metric_space_symetrie} $\metric[M](x,y) = \metric[M](y,x)$.
% \item \label{metric_space_triangle_equation} for all $x,y,z \in M$ $\metric[M](x,y) < \metric[M](x,z) + \metric[M](z,y)$ or $\metric[M](x,y) = \metric[M](x,z) + \metric[M](z,y)$.
% \item \label{metric_space_topology} $M$ is a topological space.
% \item \label{metric_space_opens} for all $x \in M$ for all $r \in \reals$ $\{z \in M \mid \metric[M](x,z) < r\} \in \opens$.
% \end{enumerate}
%\end{struct}
%\begin{definition}\label{open_ball}
% $\openball{r}{x}{M} = \{z \in M \mid \metric(x,z) < r\}$.
%\end{definition}
%\begin{proposition}\label{open_ball_is_open}
% Let $M$ be a metric space,let $r \in \reals $, let $x$ be an element of $M$.
% Then $\openball{r}{x}{M} \in \opens[M]$.
%\end{proposition}
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