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\begin{document}

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\section*{compact ordered space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{category_theory}{}\paragraph*{{$(0,1)$-Category theory}}\label{category_theory}

[[!include (0,1)-category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{categories_of_compact_ordered_spaces}{Categories of compact ordered spaces}\dotfill \pageref*{categories_of_compact_ordered_spaces} \linebreak
\noindent\hyperlink{as_algebras}{As algebras}\dotfill \pageref*{as_algebras} \linebreak
\noindent\hyperlink{connection_with_stably_compact_spaces}{Connection with stably compact spaces}\dotfill \pageref*{connection_with_stably_compact_spaces} \linebreak
\noindent\hyperlink{see_also}{See also}\dotfill \pageref*{see_also} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea}

Compact ordered spaces can be thought of as a refinement of the notion of [[compact Hausdorff space]] to [[order theory|ordered]] spaces. They share many properties of compact Hausdorff spaces, where equality is replaced by the order relation. These spaces are a convenient choice for notions of [[convergence]] in an [[order theory|ordered]] setting.

In particular, just as [[compact Hausdorff spaces]] can be seen as [[algebra over a monad|algebras]] of the [[ultrafilter\#ultrafilters\_form\_a\_monad|ultrafilter monad]] on [[Set]], compact ordered spaces can be seen as algebras of the [[prime upper filter monad]] on [[Pos]], (see \hyperlink{as_algebras}{below}).

The concept was introduced by Nachbin (see \hyperlink{nachbin}{Nachbin `65}).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{compact ordered space} or \textbf{compact pospace} is a [[compact topological space]] $X$ equipped with a [[partial order]] which is [[closed subset|closed]] as a [[subset]] of $X\times X$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{itemize}%
\item A compact ordered space is always [[Hausdorff]], i.e. it is a [[compactum]]. To see this, note that since the [[relation]] is a closed subset of $X\times X$, so is the [[opposite relation]], and hence their intersection too, which is the [[diagonal]] of $X\times X$.


\item Conversely, a [[compact Hausdorff space]] can be seen as a compact ordered space with the [[discrete order]].


\item In a compact ordered space, the [[up-sets]] and [[down-sets]] $\uparrow\!\{x\}$ and $\downarrow\!\{x\}$ are always [[closed set|closed]] and [[compact]].



\end{itemize}
(\ldots{})

\hypertarget{categories_of_compact_ordered_spaces}{}\subsection*{{Categories of compact ordered spaces}}\label{categories_of_compact_ordered_spaces}

A canonical choice of [[morphisms]] between compact ordered spaces is [[continuous map|continuous]], [[monotone maps]], which form a category. This category is usually just called the \textbf{category of compact ordered spaces}, and denoted by \textbf{$CompOrd$}.

With the 2-cell structure given by the [[pointwise order]], \textbf{$CompOrd$} becomes a [[locally posetal 2-category]].

(\ldots{})

\hypertarget{as_algebras}{}\subsection*{{As algebras}}\label{as_algebras}

For now, see \hyperlink{flagg}{Flagg `96}.

(\ldots{})

\hypertarget{connection_with_stably_compact_spaces}{}\subsection*{{Connection with stably compact spaces}}\label{connection_with_stably_compact_spaces}

For now, see \hyperlink{jung2}{Jung `04}.

(\ldots{})

\hypertarget{see_also}{}\subsection*{{See also}}\label{see_also}

\begin{itemize}%
\item [[compact Hausdorff space]]
\item [[stably compact space]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Leopoldo Nachbin, \emph{Topology and Order}, Van Nostrand, 1965.


\item Bob Flagg, \emph{Algebraic theories of compact pospaces}, Topology and its Applications, 1996 (\href{https://doi.org/10.1016/S0166-8641%2896%2900117-4}{doi:10.1016/S0166-8641(96)00117-4}).


\item Achim Jung, \emph{Stably compact spaces and the probabilistic powerspace construction}, ENTCS 87, 2004 (\href{https://doi.org/10.1016/j.entcs.2004.10.001}{doi:10.1016/j.entcs.2004.10.001}).



\end{itemize}
[[!redirects compact ordered spaces]] [[!redirects ordered compact space]] [[!redirects ordered compact spaces]] [[!redirects compact ordered topological space]] [[!redirects compact ordered topological spaces]] [[!redirects ordered compact topological space]] [[!redirects ordered compact topological spaces]] [[!redirects compact pospace]] [[!redirects compact pospaces]] [[!redirects CompOrd]] [[!redirects category of compact ordered spaces]] [[!redirects category of ordered compact spaces]] [[!redirects category of compact ordered topological spaces]] [[!redirects category of ordered compact topological spaces]] [[!redirects category of compact ordered spaces]] [[!redirects category of compact pospaces]]



\end{document}