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/order-topology.tex | 32 +++++++++++++++++++++++++++++---
 1 file changed, 29 insertions(+), 3 deletions(-)

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

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}
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