From cfd5061ced34f061e84ecca2a266f8f4cd01ce36 Mon Sep 17 00:00:00 2001
From: Simon-Kor <52245124+Simon-Kor@users.noreply.github.com>
Date: Tue, 30 Apr 2024 12:26:13 +0200
Subject: Adding the first formalisation of reals

---
 library/topology/order-topology.tex | 7 +++++++
 1 file changed, 7 insertions(+)
 create mode 100644 library/topology/order-topology.tex

(limited to 'library/topology/order-topology.tex')

diff --git a/library/topology/order-topology.tex b/library/topology/order-topology.tex
new file mode 100644
index 0000000..afa8755
--- /dev/null
+++ b/library/topology/order-topology.tex
@@ -0,0 +1,7 @@
+\import{topology/topological-space.tex}
+
+\section{Order Topology}
+
+\begin{definition}
+    A 
+\end{definition}
-- 
cgit v1.2.3


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/numbers.tex                 | 72 +++++++++++++++++----------------
 library/topology/metric-space.tex   | 80 +++++++++++++++++++++++++++++++++++++
 library/topology/order-topology.tex | 32 +++++++++++++--
 3 files changed, 146 insertions(+), 38 deletions(-)
 create mode 100644 library/topology/metric-space.tex

(limited to 'library/topology/order-topology.tex')

diff --git a/library/numbers.tex b/library/numbers.tex
index 93623fa..afb7d3f 100644
--- a/library/numbers.tex
+++ b/library/numbers.tex
@@ -4,6 +4,14 @@
 
 \section{The real numbers}
 
+%TODO: Implementing Notion for negativ number such as -x.
+
+%TODO:
+%\inv{} für inverse benutzen. Per Signatur einfüheren und dann axiomatisch absicher
+%\cdot für multiklikation verwenden. 
+%< für die relation benutzen.
+
+
 \begin{signature}
     $\reals$ is a set.
 \end{signature}
@@ -22,19 +30,31 @@
 
 \begin{axiom}\label{reals_axiom_order}
     $\lt[\reals]$ is an order on $\reals$.
-    %$\reals$ is an ordered set.
 \end{axiom}
 
 \begin{axiom}\label{reals_axiom_strictorder}
     $\lt[\reals]$ is a strict order.
 \end{axiom}
 
+\begin{abbreviation}\label{less_on_reals}
+    $x < y$ iff $(x,y) \in \lt[\reals]$.
+\end{abbreviation}
+
+\begin{abbreviation}\label{greater_on_reals}
+    $x > y$ iff $y < x$.
+\end{abbreviation}
+
+\begin{abbreviation}\label{lesseq_on_reals}
+    $x \leq y$ iff it is wrong that $x > y$.
+\end{abbreviation}
+
+\begin{abbreviation}\label{greatereq_on_reals}
+    $x \geq y$ iff it is wrong that $x < y$.
+\end{abbreviation}
+
 \begin{axiom}\label{reals_axiom_dense}
     For all $x,y \in \reals$ if $(x,y)\in \lt[\reals]$ then 
     there exist $z \in \reals$ such that $(x,z) \in \lt[\reals]$ and $(z,y) \in \lt[\reals]$.
-    
-    %For all $X,Y \subseteq \reals$ if for all $x,y$ $x\in X$ and $y \in Y$ such that $x \lt[\reals] y$ 
-    %then there exist a $z \in \reals$ such that if $x \neq z$ and $y \neq z$ $x \lt[\reals] z$ and $z \lt[\reals] y$.
 \end{axiom}
 
 \begin{axiom}\label{reals_axiom_order_def}
@@ -57,18 +77,12 @@
 \begin{axiom}\label{reals_axiom_zero_in_reals}
     $\zero \in \reals$.
 \end{axiom}
-
-%\begin{axiom}\label{reals_axiom_one_in_reals}
-%    $\one \in \reals$.
-%\end{axiom}
-    
+  
 \begin{axiom}\label{reals_axiom_zero}
-    %There exist $\zero \in \reals$ such that 
     For all $x \in \reals$ $x + \zero = x$. 
 \end{axiom}
 
 \begin{axiom}\label{reals_axiom_one}
-    %There exist $1 \in \reals$ such that 
     For all $x \in \reals$ $1 \neq \zero$ and $x \times 1 = x$.
 \end{axiom}
 
@@ -76,11 +90,6 @@
     For all $x \in \reals$ there exist $y \in \reals$ such that $x + y = \zero$.
 \end{axiom}
 
-%TODO: Implementing Notion for negativ number such as -x.
-
-%\begin{abbreviation}\label{reals_notion_minus}
-%    $y = -x$ iff $x + y = \zero$.
-%\end{abbreviation}                         %This abbrevation result in a killed process.
 
 \begin{axiom}\label{reals_axiom_mul_invers}
     For all $x \in \reals$ there exist $y \in \reals$ such that  $x \neq \zero$ and $x \times y = 1$.
@@ -98,27 +107,8 @@
     For all $x,y,z \in \reals$ if $x + y = x + z$ then $y = z$.
 \end{proposition}
 
-\begin{signature}
-    $\invers$ is a set.
-\begin{signature}
-
-%TODO:
-%x \rless y in einer signatur hinzufügen und  dann axiom x+z = y und dann \rlt in def per iff
-%\inv{} für inverse benutzen. Per Signatur einfüheren und dann axiomatisch absicher
-%\cdot für multiklikation verwenden. 
-%< für die relation benutzen.
-
-%\begin{signature}
-%    $y^{\rightarrow}$ is a function.
-%\end{signature}
 
-%\begin{axiom}\label{notion_multi_invers}
-%    If $y \in \reals$ then $\invers{y} \in \reals$ and $y \times y^{\rightarrow} = 1$.
-%\end{axiom}
 
-%\begin{abbreviation}\label{notion_fraction}
-%    $\frac{x}{y} = x \times y^{\rightarrow}$.
-%\end{abbreviation}
 
 \begin{lemma}\label{order_reals_lemma1}
     For all $x,y,z \in \reals$ such that $(\zero,x) \in \lt[\reals]$ 
@@ -138,3 +128,15 @@
     if $(y,z) \in \lt[\reals]$ 
     then $((x \times z), (x \times y)) \in \lt[\reals]$.
 \end{lemma}
+
+\begin{lemma}\label{a}
+    For all $x,y \in \reals$ if $x > y$ then $x \geq y$.
+\end{lemma}
+
+\begin{lemma}\label{aa}
+    For all $x,y \in \reals$ if $x < y$ then $x \leq y$.
+\end{lemma}
+
+\begin{lemma}\label{aaa}
+    For all $x,y \in \reals$ if $x \leq y \leq x$ then $x=y$.
+\end{lemma}
\ No newline at end of file
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}
+
diff --git a/library/topology/order-topology.tex b/library/topology/order-topology.tex
index afa8755..2dd026d 100644
--- a/library/topology/order-topology.tex
+++ b/library/topology/order-topology.tex
@@ -1,7 +1,33 @@
 \import{topology/topological-space.tex}
+\import{order/order.tex}
 
 \section{Order Topology}
 
-\begin{definition}
-    A 
-\end{definition}
+\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}
-- 
cgit v1.2.3

