\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*{formal topology}

\begin{quote}%
This entry is about a generalized notion of [[topology]]. For the notion of \emph{formal space} in the sense of [[rational homotopy theory]], see [[formal dg-algebra]].


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

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

[[!include topology - contents]]

\hypertarget{formal_topology}{}\section*{{Formal topology}}\label{formal_topology}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Formal topology is a programme for doing [[topology]] in a [[finite mathematics|finite]], [[predicative mathematics|predicative]], and [[constructive mathematics|constructive]] fashion.

It is a kind of [[pointless topology]]; in the context of [[classical mathematics]], it reproduces the theory of [[locales]] rather than [[topological spaces]] (although of course one can recover topological spaces from locales).

The basic definitions can be motivated by an attempt to study locales entirely through the [[posites]] that generate them. However, in order to recover all basic topological notions (particularly those associated with \emph{closed} rather than \emph{open} features) predicatively, we need to add a `positivity' relation to the `coverage' relation of sites.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

A \textbf{formal topology} or \textbf{formal space} is a [[set]] $S$ together with

\begin{itemize}%
\item an [[element]] $\top$ of $S$,
\item a [[binary operation]] $\cap$ on $S$,
\item a [[binary relation]] $\lhd$ between elements of $S$ and [[subsets]] of $S$, and
\item a [[unary relation]] $\Diamond$ on $S$,

\end{itemize}
such that

\begin{enumerate}%
\item $a = b$ whenever $a \lhd \{b\}$ and $b \lhd \{a\}$,
\item $a \lhd U$ whenever $a \in U$,
\item $a \lhd V$ whenever $a \lhd U$ and $x \lhd V$ for all $x \in U$,
\item $a \cap b \lhd U$ whenever $a \lhd U$ or $b \lhd U$,
\item $a \lhd \{ x \cap y \;|\; x \in U,\; y \in V \}$ whenever $a \lhd U$ and $a \lhd V$,
\item $a \lhd \{\top\}$,
\item $\Diamond x$ for some $x \in U$ whenever $\Diamond a$ and $a \lhd U$, and
\item $a \lhd U$ whenever $a \lhd U$ if $\Diamond a$,

\end{enumerate}
for all $a$, $b$, $U$, and $V$.

We interpret the elements of $S$ as \textbf{basic opens} in the formal space. We call $\top$ the \textbf{[[improper subset|entire space]]} and $a \cap b$ the \textbf{[[intersection]]} of $a$ and $b$. We say that $a$ is \textbf{covered} by $U$ or that $U$ is a \textbf{[[cover]]} of $a$ if $a \lhd U$. We say that $a$ is \textbf{positive} or \textbf{[[inhabited subset|inhabited]]} if $\Diamond a$. (For a [[topological space]] equipped with a strict [[topological base]] $S$, taking these intepretations literally does in fact define a formal space; see the Examples.)

Some immediate points to notice:

\begin{itemize}%
\item If we drop (1), then the hypothesis of (1) defines an [[equivalence relation]] on $S$ which is a [[congruence]] for $\top$, $\cap$, $\lhd$, and $\Diamond$, so that we may simply pass to the [[quotient set]]. In appropriate foundations, we can even allow $S$ to be a [[preset]] originally, then use (1) as a definition of equality.
\item We can prove that $(S,\cap,\top)$ is a bounded [[semilattice]]; if (as the notation suggests) we interpret this as a [[meet]]-semilattice, then $a \leq b$ if and only if $a \lhd \{b\}$. Conversely, we could require that $(S,\cap,\top)$ be a semilattice originally, then let (1) say that $a \leq b$ whenever $a \lhd \{b\}$.
\item We can prove that $\Diamond a$ holds iff every cover of $a$ is inhabited and that $\Diamond a$ fails iff $a \lhd \empty$. Accordingly, this predicate is uniquely definable (in two equivalent ways, one impredicative and one nonconstructive) in a classical treatment; only in a treatment that is both predicative and constructive do we need to include it in the axioms. See [[positivity predicate]].

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

Let $X$ be a [[topological space]], and let $S$ be the collection of [[open subsets]] of $X$. Let $\top$ be $X$ itself, and let $a \cap b$ be the literal intersection of $a$ and $b$ for $a, b \in S$. Let $a \lhd U$ if and only $U$ is literally an [[open cover]] of $a$, and let $\Diamond a$ if and only if $a$ is literally inhabited. Then $(S,\top,\cap,\lhd,\Diamond)$ is a formal topology.

The above example is impredicative (since the collection of open subsets is generally large), but now let $S$ be a [[base for the topology]] of $X$ which is strict in the sense that it is closed under finitary [[intersection]]. Let the other definitions be as before. Then $(S,\top,\cap,\lhd,\Diamond)$ is a formal topology.

More generally, let $B$ be a [[subbase]] for the topology of $X$, and let $S$ be the [[free monoid]] on $B$, that is the set of [[finite lists]] of elements of $B$ (so this example is not strictly finitist), modulo the [[equivalence relation]] by which two lists are identified if their [[intersections]] are equal. Let $\top$ be the [[empty list]], let $a \cap b$ be the [[concatenation]] of $a$ and $b$, let $a \lhd U$ if the intersection of $a$ is contained in the [[union]] of the intersections of the elements of $U$, and let $\Diamond a$ if the intersection of $a$ is inhabited. Then $(S,\top,\cap,\lhd,\Diamond)$ is a formal topology.

Let $X$ be an [[accessible locale]] generated by a [[posite]] whose underlying [[poset]] $S$ is a (meet)-[[semilattice]]. Let $\top$ and $\cap$ be as in the semilattice structure on $S$, and let $a \lhd U$ if $U$ contains a basic cover (in the posite structure on $S$) of $a$. Then we get a formal topology, defining $\Diamond$ in the unique way.

The last example is not predicative, and this is in part \emph{why} one studies formal topologies instead of sites, if one wishes to be strictly predicative. (It still needs to be motivated that we want $\Diamond$ at all.)

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

\begin{itemize}%
\item [[Mike Fourman]] and Grayson (1982); \emph{Formal Spaces}. This is the original development, intended as an application of locale theory to logic.


\item [[Giovanni Sambin]] (1987); \emph{Intuitionistic formal spaces}; \href{http://www.math.unipd.it/~sambin/txt/ifs87-97.pdf}{pdf}.

\begin{itemize}%
\item This is the probably the main reference on the subject.
\item Warning: you can ignore the material about [[foundations]] and [[type theory]], but if you do read any of it, the term `category' means (roughly) [[proper class]]; this was common in type theory in the 1980s.

\end{itemize}

\item [[Giovanni Sambin]] (2001); \emph{Some points in formal topology}; \href{http://www.math.unipd.it/~sambin/txt/SP.pdf}{pdf}. This has newer results, alternative formulations, etc.


\item [[Erik Palmgren]], \emph{From Intuitionistic to Point-Free Topology: On the Foundation of Homotopy Theory}, Logicism, Intuitionism, and Formalism Volume 341 of the series Synthese Library pp 237-253, 2005 (\href{http://www2.math.uu.se/~palmgren/homotopy_rev2.pdf}{pdf})



\end{itemize}
[[!redirects formal topology]] [[!redirects formal topologies]]

[[!redirects formal space]] [[!redirects formal spaces]] [[!redirects formal topological space]] [[!redirects formal topological spaces]]



\end{document}