Merge branch 'main' into main

3 files changed, 254 insertions, 0 deletions

diff --git a/library/topology/basis.tex b/library/topology/basis.tex
index d8cfeaf..15910f9 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}
 
@@ -48,3 +50,70 @@
     $\genOpens{B}{X} = \left\{ U\in\pow{X} \middle| \textbox{for all $x\in U$ there exists $V\in B$
     \\ such that $x\in V\subseteq U$}\right\}$.
 \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
new file mode 100644
index 0000000..1c6a0ca
--- /dev/null
+++ b/library/topology/metric-space.tex
@@ -0,0 +1,152 @@
+\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 $\reals$ 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{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{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 $x \in M$ 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{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}
+
+
+
+
+%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{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
+%    \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}
+
+
+
+
+
+
+%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.
+
diff --git a/library/topology/order-topology.tex b/library/topology/order-topology.tex
new file mode 100644
index 0000000..2dd026d
--- /dev/null
+++ b/library/topology/order-topology.tex
@@ -0,0 +1,33 @@
+\import{topology/topological-space.tex}
+\import{order/order.tex}
+
+\section{Order Topology}
+
+\begin{abbreviation}\label{open_interval}
+    $z \in \oointervalof{x}{y}$ iff $x \mathrel{R} y$ and $x \mathrel{R} z$ and $z \mathrel{R} y$.
+    %$\oointervalof{x}{y}{X} = \{ z \mid x \in X, y \in X, z \in X x \mathrel{R} y \wedge x \mathrel{R} z \wedge z \mathrel{R} y\}$. 
+\end{abbreviation}
+
+\begin{struct}\label{order_topology}
+    A ordertopology space $X$ is a onesorted structure equipped with
+    \begin{enumerate}
+        \item $<$
+    \end{enumerate}
+    such that 
+    \begin{enumerate}
+        \item \label{order_topology_1} $<$ is a strict order on $X$
+        \item \label{order_topology_2}  
+        \item \label{order_topology_3}
+        \item \label{order_topology_4}
+        \item \label{order_topology}
+        \item \label{order_topology}
+        \item \label{order_topology}
+    \end{enumerate}
+\end{struct}
+
+
+
+%\begin{definition}\label{order_topology}
+%    $X$ has the order topology iff for all $x,y \in X$ $X$ has a strict order $R$ and $\oointervalof{x}{y}{X} \in \opens[X]$ and $X$ is a topological space.
+%    %$O$ is the order Topology on $X$ iff for all $x,y \in X$ $X$ has a strict order $R$ and $(x,y) \in O$ and $O$ is . 
+%\end{definition}