From 3795588d157864a411baf2fc3afb31f9f5184d93 Mon Sep 17 00:00:00 2001
From: Simon-Kor <52245124+Simon-Kor@users.noreply.github.com>
Date: Tue, 7 May 2024 14:41:15 +0200
Subject: Formalization of metric spaces and some cleaning of numbers.tex

Formalization of metric spaces:
Therefore we introduced the predicate metric and its axiomatization.
Then we introduced the term metric space in dependence of a metric function.
This metric space is automatically a a topological space.
---
 library/topology/metric-space.tex | 80 +++++++++++++++++++++++++++++++++++++++
 1 file changed, 80 insertions(+)
 create mode 100644 library/topology/metric-space.tex

(limited to 'library/topology/metric-space.tex')

diff --git a/library/topology/metric-space.tex b/library/topology/metric-space.tex
new file mode 100644
index 0000000..7021a60
--- /dev/null
+++ b/library/topology/metric-space.tex
@@ -0,0 +1,80 @@
+\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}
+
-- 
cgit v1.2.3


From 08019dcdaf3b13bb8ce554dfd5377690bb508c6d Mon Sep 17 00:00:00 2001
From: Simon-Kor <52245124+Simon-Kor@users.noreply.github.com>
Date: Tue, 7 May 2024 18:08:04 +0200
Subject: formalisation mertic optimized

---
 library/numbers.tex               | 18 ++++++++++-------
 library/topology/metric-space.tex | 41 ++++++++++++++++++++++++++-------------
 2 files changed, 38 insertions(+), 21 deletions(-)

(limited to 'library/topology/metric-space.tex')

diff --git a/library/numbers.tex b/library/numbers.tex
index df47d81..a0e2211 100644
--- a/library/numbers.tex
+++ b/library/numbers.tex
@@ -10,7 +10,7 @@
 %\inv{} für inverse benutzen. Per Signatur einfüheren und dann axiomatisch absicher
 %\cdot für multiklikation verwenden. 
 %< für die relation benutzen.
-
+% sup und inf einfügen
 
 \begin{signature}
     $\reals$ is a set.
@@ -92,7 +92,7 @@
 
 
 \begin{axiom}\label{reals_axiom_mul_invers}
-    For all $x \in \reals$ there exist $y \in \reals$ such that  $x \neq \zero$ and $x \times y = 1$.
+    For all $x \in \reals$ such that $x \neq \zero$ there exist $y \in \reals$ such that $x \times y = 1$.
 \end{axiom}
 
 \begin{axiom}\label{reals_axiom_disstro1}
@@ -107,7 +107,10 @@
     For all $x,y,z \in \reals$ if $x + y = x + z$ then $y = z$.
 \end{proposition}
 
-
+\begin{axiom}\label{reals_axiom_dedekind_complete}
+    For all $X,Y,x,y$ such that $X,Y \subseteq \reals$ and $x \in X$ and $y \in Y$ and $x < y$ we have there exist $z \in \reals$
+    such that $x < z < y$.
+\end{axiom}
 
 
 \begin{lemma}\label{order_reals_lemma1}
@@ -129,14 +132,15 @@
     then $(x \times z) < (x \times y)$.
 \end{lemma}
 
-\begin{lemma}\label{a}
+\begin{lemma}\label{o4rder_reals_lemma}
     For all $x,y \in \reals$ if $x > y$ then $x \geq y$.
 \end{lemma}
 
-\begin{lemma}\label{aa}
+\begin{lemma}\label{order_reals_lemma5}
     For all $x,y \in \reals$ if $x < y$ then $x \leq y$.
 \end{lemma}
 
-\begin{lemma}\label{aaa}
+\begin{lemma}\label{order_reals_lemma6}
     For all $x,y \in \reals$ if $x \leq y \leq x$ then $x=y$.
-\end{lemma}
\ No newline at end of file
+\end{lemma}
+
diff --git a/library/topology/metric-space.tex b/library/topology/metric-space.tex
index 7021a60..2a31d95 100644
--- a/library/topology/metric-space.tex
+++ b/library/topology/metric-space.tex
@@ -4,23 +4,22 @@
 
 \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 
+\begin{definition}\label{metric}
+    $f$ is a metric on $M$ iff $f$ is a function from $M \times M$ to $\reals$ 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{axiom}
+\end{definition}
 
 \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 $f(x,z)<r$ } \}$.
+    $\openball{r}{x}{f} = \{z \in M \mid \text{ $f$ is a metric on $M$ and $f(x,z) < r$ } \}$.
 \end{definition}
 
+%TODO: \metric_opens{d} =  {hier die construction für topology}
+%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
@@ -29,8 +28,7 @@
     \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_metric}                   $\metric[M]$ is a metric on $M$.
         \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}
@@ -45,12 +43,27 @@
 \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 $M$.
+    Let $M$ be a metric space.
+    For all $U,V \subseteq M$ if $U$ is an open ball in $M$ and $V$ is an open ball in $M$ then $U \union V$ is open in $M$.
 \end{lemma}
 
 
+%\begin{definition}\label{lenght_of_interval} %TODO: take minus if its implemented
+%    $\lenghtinterval{x}{y} = r$ 
+%\end{definition}
+
+
+
+
+
+\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}
+
+
+
+
 
 %\begin{struct}\label{metric_space}  
 %    A metric space $M$ is a onesorted structure equipped with
-- 
cgit v1.2.3


From 63518b2e0bfdf0308fba30920a7a3bb7f61da994 Mon Sep 17 00:00:00 2001
From: Simon-Kor <52245124+Simon-Kor@users.noreply.github.com>
Date: Tue, 14 May 2024 16:55:40 +0200
Subject: work on metric spaces

---
 library/numbers.tex               | 49 +++++++++++++++++++++++++++++++++++++++
 library/topology/metric-space.tex | 45 +++++++++++++++++++++++++++++++++--
 2 files changed, 92 insertions(+), 2 deletions(-)

(limited to 'library/topology/metric-space.tex')

diff --git a/library/numbers.tex b/library/numbers.tex
index a0e2211..5dd06da 100644
--- a/library/numbers.tex
+++ b/library/numbers.tex
@@ -144,3 +144,52 @@
     For all $x,y \in \reals$ if $x \leq y \leq x$ then $x=y$.
 \end{lemma}
 
+\begin{axiom}\label{reals_axiom_minus}
+    For all $x \in \reals$ $x - x = \zero$.
+\end{axiom}
+
+\begin{lemma}\label{reals_minus}
+    Assume $x,y \in \reals$. If $x - y = \zero$ then $x=y$.
+\end{lemma}
+
+%\begin{definition}\label{reasl_supremum} %expaction "there exists" after \mid
+%    $\rsup{X} = \{z \mid \text{ $z \in \reals$ and for all $x,y$ such that $x \in X$ and $y,x \in \reals$ and $x < y$ we have $z \leq y$ }\}$.
+%\end{definition}
+
+\begin{definition}\label{upper_bound}
+    $x$ is an upper bound of $X$ iff for all $y \in X$ we have $x > y$.
+\end{definition}
+
+\begin{definition}\label{least_upper_bound}
+    $x$ is a least upper bound of $X$ iff $x$ is an upper bound of $X$ and for all $y$ such that $y$ is an upper bound of $X$ we have $x \leq y$.
+\end{definition}
+
+\begin{lemma}\label{supremum_unique}
+    %Let $x,y \in \reals$ and let $X$ be a subset of $\reals$.
+    If $x$ is a least upper bound of $X$ and $y$ is a least upper bound of $X$ then $x = y$.
+\end{lemma}
+
+\begin{definition}\label{supremum_reals}
+    $x$ is the supremum of $X$ iff $x$ is a least upper bound of $X$.
+\end{definition}
+
+
+
+
+\begin{definition}\label{lower_bound}
+    $x$ is an lower bound of $X$ iff for all $y \in X$ we have $x < y$.
+\end{definition}
+
+\begin{definition}\label{greatest_lower_bound}
+    $x$ is a greatest lower bound of $X$ iff $x$ is an lower bound of $X$ and for all $y$ such that $y$ is an lower bound of $X$ we have $x \geq y$.
+\end{definition}
+
+\begin{lemma}\label{infimum_unique}
+    %Let $x,y \in \reals$ and let $X$ be a subset of $\reals$.
+    If $x$ is a greatest lower bound of $X$ and $y$ is a greatest lower bound of $X$ then $x = y$.
+\end{lemma}
+
+\begin{definition}\label{infimum_reals}
+    $x$ is the supremum of $X$ iff $x$ is a greatest lower bound of $X$.
+\end{definition}
+
diff --git a/library/topology/metric-space.tex b/library/topology/metric-space.tex
index 2a31d95..8ec83f7 100644
--- a/library/topology/metric-space.tex
+++ b/library/topology/metric-space.tex
@@ -1,6 +1,7 @@
 \import{topology/topological-space.tex}
 \import{numbers.tex}
 \import{function.tex}
+\import{set/powerset.tex}
 
 \section{Metric Spaces}
 
@@ -17,7 +18,46 @@
     $\openball{r}{x}{f} = \{z \in M \mid \text{ $f$ is a metric on $M$ and $f(x,z) < r$ } \}$.
 \end{definition}
 
-%TODO: \metric_opens{d} =  {hier die construction für topology}
+
+
+\begin{definition}\label{induced_topology}
+    $O$ is the induced topology of $d$ in $M$ iff 
+    $O \subseteq \pow{M}$ and
+    $d$ is a metric on $M$ and 
+    for all $x,r,A,B,C$ 
+        such that $x \in M$ and $r \in \reals$ and $A,B \in O$ and $C$ is a family of subsets of $O$
+        we have $\openball{r}{x}{d} \in O$ and $\unions{C} \in O$ and $A \inter B \in O$.
+\end{definition}
+
+%\begin{definition}
+%    $\projcetfirst{A} = \{a \mid \exists x \in X \text{there exist $x \i }  \}$
+%\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 $x \in M$ we have $O = \openball{r}{x}{d}$ } \}$.
+\end{definition}
+
+
+\begin{definition}\label{toindsas}
+    $\metricopens{d}{M} = \{O \in \pow{M} \mid \text{
+        $d$ is a metric on $M$ and
+        for all $x,r,A,B,C$ 
+        such that $x \in M$ and $r \in \reals$ and $A,B \in O$ and $C$ is a family of subsets of $O$
+        we have $\openball{r}{x}{d} \in O$ and $\unions{C} \in O$ and $A \inter B \in O$.
+    }  \}$.
+    
+\end{definition}
+
+\begin{theorem}\label{metric_induce_a_topology}
+    
+
+
+\end{theorem}
+
+
+
+
+%TODO: \metric_opens{d} =  {hier die construction für topology} DONE.
 %TODO: Die induzierte topology definieren und dann in struct verwenden.
 
 
@@ -44,7 +84,7 @@
 
 \begin{lemma}\label{union_of_open_balls_is_open}
     Let $M$ be a metric space.
-    For all $U,V \subseteq M$ if $U$ is an open ball in $M$ and $V$ is an open ball in $M$ then $U \union V$ is open in $M$.
+    For all $U,V \subseteq M$ if $U$, $V$ are open balls in $M$ then $U \union V$ is open in $M$.
 \end{lemma}
 
 
@@ -56,6 +96,7 @@
 
 
 
+
 \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.
-- 
cgit v1.2.3


From ecfb1a66f2159e078199e54edf8a80004c28195a Mon Sep 17 00:00:00 2001
From: Simon-Kor <52245124+Simon-Kor@users.noreply.github.com>
Date: Tue, 28 May 2024 15:33:45 +0200
Subject: proofing some lammes about topological basis

---
 .gitignore                        |  2 ++
 library/topology/basis.tex        | 75 +++++++++++++++++++++++++++++++++++++--
 library/topology/metric-space.tex | 60 ++++++++++++++++++++-----------
 3 files changed, 113 insertions(+), 24 deletions(-)

(limited to 'library/topology/metric-space.tex')

diff --git a/.gitignore b/.gitignore
index ddb98c3..49c3120 100644
--- a/.gitignore
+++ b/.gitignore
@@ -42,3 +42,5 @@ haddocks/
 stack.yaml.lock
 zf*.svg
 Anmerkungen.txt
+vampire-taks-with-not-expacted-behavoir/
+
diff --git a/library/topology/basis.tex b/library/topology/basis.tex
index bca42f0..61a358f 100644
--- a/library/topology/basis.tex
+++ b/library/topology/basis.tex
@@ -1,4 +1,6 @@
 \import{topology/topological-space.tex}
+\import{set.tex}
+\import{set/powerset.tex}
 
 \subsection{Topological basis}
 
@@ -44,7 +46,74 @@
     there exists $W\in B$ such that $x\in W\subseteq U, V$.
 \end{definition}
 
-\begin{definition}\label{genOpens}
-    $\genOpens{B}{X} = \{ U\in\pow{X} \mid for all $x\in U$ there exists $V\in B$
-    such that $x\in V\subseteq U$\}$.
+\begin{definition}\label{genopens}
+    $\genOpens{B}{X} = \{ U\in\pow{X} \mid \text{for all $x\in U$ there exists $V\in B$
+    such that $x\in V\subseteq U$} \}$.
 \end{definition}
+
+\begin{lemma}\label{emptyset_in_genopens}
+    Assume $B$ is a topological basis for $X$.
+    $\emptyset \in \genOpens{B}{X}$.
+\end{lemma}
+
+\begin{lemma}\label{all_is_in_genopens}
+    Assume $B$ is a topological basis for $X$.
+    $X \in \genOpens{B}{X}$.
+\end{lemma}
+\begin{proof}
+    $B$ covers $X$ by \cref{topological_prebasis_iff_covering_family,topological_basis}.
+    $\unions{B} \in \genOpens{B}{X}$.
+    $X \subseteq \unions{B}$.
+\end{proof}
+
+\begin{lemma}\label{union_in_genopens}
+    Assume $B$ is a topological basis for $X$.
+    For all $F\subseteq \genOpens{B}{X}$ we have $\unions{F}\in\genOpens{B}{X}$.
+\end{lemma}
+\begin{proof}
+    Omitted.
+\end{proof}
+
+
+
+
+\begin{lemma}\label{inters_in_genopens}
+    Assume $B$ is a topological basis for $X$.
+    %For all $A, C$ 
+    If $A\in \genOpens{B}{X}$ and $C\in \genOpens{B}{X}$ then $(A\inter C) \in \genOpens{B}{X}$.
+\end{lemma}
+\begin{proof}
+    
+    Show $(A \inter C) \in \pow{X}$.
+    \begin{subproof}
+        $(A \inter C) \subseteq X$ by assumption.
+    \end{subproof}
+    Therefore for all $A, C \in \genOpens{B}{X}$ we have $(A \inter C) \in \pow{X}$.
+    
+
+    Show for all $x\in (A\inter C)$ there exists $W \in B$
+    such that $x\in W$ and $W \subseteq (A\inter C)$.
+    \begin{subproof}
+        Fix $x \in (A\inter C)$.
+        There exist $V' \in B$ such that $x \in V'$ and $V' \subseteq A$ by assumption.  %TODO: Warum muss hier by assumtion hin?
+        There exist $V'' \in B$ such that $x \in V''$ and $V'' \subseteq C$ by assumption.
+        There exist $W \in B$ such that $x \in W$ and $W \subseteq v'$ and $W \subseteq V''$ by assumption.
+        
+        Show $W \subseteq (A\inter C)$.
+        \begin{subproof}
+            %$W \subseteq v'$ and $W \subseteq V''$.
+            For all $y \in W$ we have $y \in V'$ and $y \in V''$ by assumption.
+        \end{subproof}
+    \end{subproof}
+    %Therefore for all $A, C, x$ such that $A \in \genOpens{B}{X}$ and $C \in \genOpens{B}{X}$ and $x \in (A \inter C)$ we have there exists $W \in B$
+    %such that $x\in W$ and $W \subseteq (A\inter C)$.
+
+    $(A\inter C) \in \genOpens{B}{X}$ by assumption.
+    
+ 
+\end{proof}
+
+
+
+
+
diff --git a/library/topology/metric-space.tex b/library/topology/metric-space.tex
index 8ec83f7..1c6a0ca 100644
--- a/library/topology/metric-space.tex
+++ b/library/topology/metric-space.tex
@@ -2,6 +2,7 @@
 \import{numbers.tex}
 \import{function.tex}
 \import{set/powerset.tex}
+\import{topology/basis.tex}
 
 \section{Metric Spaces}
 
@@ -20,38 +21,42 @@
 
 
 
-\begin{definition}\label{induced_topology}
-    $O$ is the induced topology of $d$ in $M$ iff 
-    $O \subseteq \pow{M}$ and
-    $d$ is a metric on $M$ and 
-    for all $x,r,A,B,C$ 
-        such that $x \in M$ and $r \in \reals$ and $A,B \in O$ and $C$ is a family of subsets of $O$
-        we have $\openball{r}{x}{d} \in O$ and $\unions{C} \in O$ and $A \inter B \in O$.
-\end{definition}
+%\begin{definition}\label{induced_topology}
+%    $O$ is the induced topology of $d$ in $M$ iff 
+%    $O \subseteq \pow{M}$ and
+%    $d$ is a metric on $M$ and 
+%    for all $x,r,A,B,C$ 
+%        such that $x \in M$ and $r \in \reals$ and $A,B \in O$ and $C$ is a family of subsets of $O$
+%        we have $\openball{r}{x}{d} \in O$ and $\unions{C} \in O$ and $A \inter B \in O$.
+%\end{definition}
 
 %\begin{definition}
 %    $\projcetfirst{A} = \{a \mid \exists x \in X \text{there exist $x \i }  \}$
 %\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 $x \in M$ we have $O = \openball{r}{x}{d}$ } \}$.
+    $\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{toindsas}
-    $\metricopens{d}{M} = \{O \in \pow{M} \mid \text{
-        $d$ is a metric on $M$ and
-        for all $x,r,A,B,C$ 
-        such that $x \in M$ and $r \in \reals$ and $A,B \in O$ and $C$ is a family of subsets of $O$
-        we have $\openball{r}{x}{d} \in O$ and $\unions{C} \in O$ and $A \inter B \in O$.
-    }  \}$.
-    
-\end{definition}
+%\begin{definition}\label{toindsas}
+%    $\metricopens{d}{M} = \{O \in \pow{M} \mid \text{
+%        $d$ is a metric on $M$ and
+%        for all $x,r,A,B,C$ 
+%        such that $x \in M$ and $r \in \reals$ and $A,B \in O$ and $C$ is a family of subsets of $O$
+%        we have $\openball{r}{x}{d} \in O$ and $\unions{C} \in O$ and $A \inter B \in O$.
+%    }  \}$.
+%    
+%\end{definition}
 
-\begin{theorem}\label{metric_induce_a_topology}
-    
+\begin{definition}\label{metricopens}
+    $\metricopens{d}{M} = \genOpens{\balls{d}{M}}{M}$.
+\end{definition}
 
 
+\begin{theorem}
+    Let $d$ be a metric on $M$.
+    $M$ is a topological space.
 \end{theorem}
 
 
@@ -70,7 +75,7 @@
     \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}                    for all $x \in M$ for all $r \in \reals$ $\openball{r}{x}{\metric[M]} \in \opens[M]$.
+        \item \label{metric_space_opens}                    $\metricopens{ \metric[M] }{M} = \opens[M]$.
     \end{enumerate}
 \end{struct}
 
@@ -132,3 +137,16 @@
 %    Then $\openball{r}{x}{M} \in \opens[M]$.
 %\end{proposition}
 
+
+
+
+
+
+%TODO:  - Basis indudiert topology lemma
+%       - Offe Bälle sind basis
+
+% Was danach kommen soll bleibt offen, vll buch oder in proof wiki
+% Trennungsaxiom, 
+
+% Notaionen aufräumen damit das gut gemercht werden kann.
+
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