\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{specialization topology}

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

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

[[!include topology - contents]]

\hypertarget{the_specialisation_topology}{}\section*{{The specialisation topology}}\label{the_specialisation_topology}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{alexandroff_topological_spaces}{Alexandroff topological spaces}\dotfill \pageref*{alexandroff_topological_spaces} \linebreak
\noindent\hyperlink{AlexandrovLocales}{Alexandroff locales}\dotfill \pageref*{AlexandrovLocales} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \textbf{specialisation topology}, also called the \textbf{Alexandroff topology}, is a natural structure of a [[topological space]] induced on the underlying [[set]] of a [[preordered set]]. This is similar to the [[Scott topology]], which is however coarser.

Spaces with this topology, called \emph{Alexandroff spaces} and named after [[Paul Alexandroff]] (Pavel Aleksandrov), should not be confused with \emph{[[Alexandrov spaces]]} (which arise in [[differential geometry]] and are named after [[Alexander Alexandrov]]).

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

\begin{defn}
\label{}\hypertarget{}{}
Let $P$ be a [[preordered set]].

Declare a [[subset]] $A$ of $P$ to be an \emph{[[open subset]]} if it is upwards-closed. That is, if $x \leq y$ and $x \in A$, then $y \in A$.

This defines a [[topology]] on $P$, called the \textbf{specialization topology} or \textbf{Alexandroff topology}.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
One may also use the convention that the open sets are the downwards-closed subsets; this is the specialisation topology on the [[opposite category|opposite]] $P^\op$.

\end{remark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[Sierpinski space]] is the poset of [[truth values]] with the specialisation topology.

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{prop}
\label{}\hypertarget{}{}
Every [[finite topological space]] is an Alexandroff space.

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
A [[preorder]] $P$ is a [[partial order|poset]] if and only if its specialisation topology is $T_0$.

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
A [[function]] between preorders is order-preserving if and only if it is a [[continuous map]] with respect to the specialisation topology.

\end{prop}
\hypertarget{alexandroff_topological_spaces}{}\subsubsection*{{Alexandroff topological spaces}}\label{alexandroff_topological_spaces}

\begin{defn}
\label{}\hypertarget{}{}
An \textbf{Alexandroff space} is a [[topological space]] for which arbitrary (as opposed to just finite) [[intersections]] of [[open subsets]] are still open.

Write

\begin{displaymath}
AlexTop \hookrightarrow Top
\end{displaymath}
for the [[full subcategory]] of [[Top]] on the Alexandroff spaces.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
Every Alexandroff space is obtained by equipping its [[specialization order]] with the Alexandroff topology.

\end{prop}
\begin{cor}
\label{}\hypertarget{}{}
The specialization topology embeds the category $\Pros$ of preordered sets [[full subcategory|fully-faithfully]] in the category [[Top]] of topological spaces.

\begin{displaymath}
Proset \hookrightarrow Top
  \,.
\end{displaymath}
If we restrict to a [[finite set|finite]] underlying set, then the categories $\Fin\Pros$ and $\Fin\Top$ of finite prosets and [[finite topological spaces]] are [[equivalence of categories|equivalent]] in this way.

\end{cor}
\hypertarget{AlexandrovLocales}{}\subsubsection*{{Alexandroff locales}}\label{AlexandrovLocales}

\begin{defn}
\label{}\hypertarget{}{}
Write $AlexLocale$ for the non-[[full subcategory|full]] [[subcategory]] of [[Locale]] whose

\begin{itemize}%
\item [[object]]s are \textbf{Alexandroff locales}, that is locales of the form $Alex P$ for $P\in Poset$ with $Open(Alex(P)) = UpSets(P)$;


\item [[morphisms]] are those morphisms of locales $f\colon Alex P \to Alex Q$, for which the dual [[inverse image]] morphism of [[frames]] $f^*\colon UpSet(Q) \to UpSet(P)$ has a [[left adjoint]] $f_!\colon UpSet(P) \to UpSet(Q)$.



\end{itemize}
\end{defn}
This appears as (\hyperlink{Caramello}{Caramello, p. 55}).

\begin{remark}
\label{}\hypertarget{}{}
By the definition of the [[2-category]] [[Locale]] (see there), this means that $AlexPoset$ consists of those morphisms which have \emph{right} adjoints in [[Locale]].

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
The functor $Alex\colon Poset \to Locale$ factors through $AlexLocale$ and exhibits an [[equivalence of categories]]

\begin{displaymath}
Alex\colon Poset \stackrel{\simeq}{\to} AlexLocale
  \,.
\end{displaymath}
\end{prop}
This appears as (\hyperlink{Caramello}{Caramello, theorem 4.2}).

\begin{prop}
\label{}\hypertarget{}{}
The category of Alexandroff locales is equivalent to that of [[completely distributive lattice|completely distributive]] [[algebraic lattice]]s.

\end{prop}
This appears as (\hyperlink{Caramello}{Caramello, remark 4.3}).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[order topology]]


\item [[Scott topology]]



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

\begin{itemize}%
\item Wikipedia (English), \href{http://en.wikipedia.org/wiki/Alexandrov_topology}{Alexandrov topology}

\end{itemize}
The original article is

\begin{itemize}%
\item [[Paul Alexandroff]], \emph{Diskrete R\"a{}ume} (\href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=5579&option_lang=eng}{web}) (1973)

\end{itemize}
Details on Alexandroff spaces are in

\begin{itemize}%
\item F. Arenas, \emph{Alexandroff spaces}, Acta Math. Univ. Comenianae Vol. LXVIII, 1 (1999), pp. 17--25 (\href{http://www.emis.de/journals/AMUC/_vol-68/_no_1/_arenas/arenas.pdf}{pdf})


\item Timothy Speer, \emph{A Short Study of Alexandroff Spaces} (\href{http://arxiv.org/abs/0708.2136}{arXiv:0708.2136})



\end{itemize}
A useful discussion of the abstract relation between posets and Alexandroff [[locales]] is in section 4.1 of

\begin{itemize}%
\item [[Olivia Caramello]], \emph{A topos-theoretic approach to Stone-type dualities} (\href{http://arxiv.org/abs/1103.3493}{arXiv:1103.3493})

\end{itemize}
See also around page 45 in

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Stone Spaces]]}

\end{itemize}
A discussion of [[abelian sheaf cohomology]] on Alexandroff spaces is in

\begin{itemize}%
\item Morten Brun, Winfried Bruns, Tim R\"o{}mer, \emph{Cohomology of partially ordered sets and local cohomology of section rings} Advances in Mathematics 208 (2007) 210--235 (\href{http://www.home.uni-osnabrueck.de/wbruns/brunsw/pdf-article/BrunBrunsRoemer.published.pdf}{pdf})

\end{itemize}
[[!redirects specialization topology]] [[!redirects specialization topologies]] [[!redirects specialisation topology]] [[!redirects specialisation topologies]] [[!redirects Alexandroff topology]] [[!redirects Alexandroff topologies]] [[!redirects Alexandrov topology]] [[!redirects Alexandrov topologies]]

[[!redirects Alexandroff space]] [[!redirects Alexandroff spaces]]

[[!redirects Alexandroff locale]] [[!redirects Alexandroff locales]] [[!redirects Alexandrov locale]] [[!redirects Alexandrov locales]]

[[!redirects completely distributive algebraic lattice]] [[!redirects completely distributive algebraic lattices]]



\end{document}