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\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}
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