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\begin{document}

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\section*{overt space}

\hypertarget{overt_spaces}{}\section*{{Overt spaces}}\label{overt_spaces}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{motivation}{Motivation}\dotfill \pageref*{motivation} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{overt_locales}{Overt locales}\dotfill \pageref*{overt_locales} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{in_synthetic_topology}{In synthetic topology}\dotfill \pageref*{in_synthetic_topology} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

An \emph{overt} space is a space that satisfies a condition [[de Morgan duality|logically dual]] to that satisfied by a [[compact space]]. An overt space is also called \emph{open} (in French, there is only one word, `ouvert').

\hypertarget{motivation}{}\subsection*{{Motivation}}\label{motivation}

Recall that a [[topological space]] $X$ is \emph{compact} if and only if, for every other space $Y$ and any open subspace $U$ of $X \times Y$, the subspace

\begin{displaymath}
\forall_X U = \{ b\colon Y \;|\; \forall\; a\colon X,\; (a, b) \in U \}
\end{displaymath}
is open in $Y$.

Dually, a space is \emph{overt} if and only if, for every other space $Y$ and any open subspace $U$ of $X \times Y$, the subspace

\begin{displaymath}
\exists_X U = \{ b\colon Y \;|\; \exists\; a\colon X,\; (a, b) \in U \}
\end{displaymath}
is open in $Y$. (Note that the duality here is only in the logical connectives, not within the category of spaces.)

Of course, \emph{every} topological space satisfies this condition; $\exists_X U$ is the [[union]] over $a$ of the open subspaces $\{ b \;|\; (a,b) \in U \}$. However, the condition is not trivial in some frameworks, such as [[constructive mathematics|constructive]] [[locale]] theory, [[formal topology]], and [[Abstract Stone Duality]].

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

To remove it from dependence on points, we write the definition like this:

A [[space]] (topological space, locale, etc) $X$ is \textbf{overt} (or \textbf{open}) if, given any space $Y$ and any [[open subspace]] $U$ of the [[product space]] $X \times Y$, there exists an open $\exists_X U$ in $Y$ that satisfies the [[universal property]] of [[existential quantification]]:

\begin{displaymath}
\exists_X U \subseteq V \;\Leftrightarrow\; U \subseteq X \times V
\end{displaymath}
for every open $V$ in $Y$.

Note that, since we quantify over all spaces $Y$ and opens $U\subseteq X\times Y$ in this definition, whether a given space is overt may depend on precisely what `space' and `open' mean (even if $X$ is an example for both meanings). For example, if $X$ is a set with the [[discrete topology]], then it is always overt if `space' means `topological space' or `locale' with the usual meaning of `open'. On the other hand, if `space' means simply `set', but `open' refers to the synthetic notion induced by a [[dominance]], then overtness of $X$ is a nontrivial condition (and in fact, if all sets are overt in this sense then the dominance is trivial).

\hypertarget{overt_locales}{}\subsection*{{Overt locales}}\label{overt_locales}

In the case of [[locales]], it is sufficient to require the above definition only for the case $Y \coloneqq 1$ the [[point]] (unlike the ``dual'' case of compactness). In this case, when the map $\exists_X\colon Op(X) \to Op(1) = TV$ (the set or class of [[truth values]]) exists, it is a [[positivity predicate]] on $Op(X)$. The behavior of this predicate depends on the foundational assumptions:

\begin{itemize}%
\item In [[classical mathematics]], it always exists; thus classically every locale is overt.


\item In impredicative [[constructive mathematics]] (such as the [[internal logic]] of a [[topos]]), the positivity predicate can be defined, but it may not satisfy the requisite univeral property of [[adjunction|adjointness]]. Thus, constructively, not every locale is overt. However, even constructively, every [[topological locale]] is overt (so a [[sober space]] is overt regardless of whether it is viewed as a topological space or as a locale).


\item In constructive [[predicative mathematics]], a positivity predicate cannot be defined and so must be given as a structure on (predicative data that generate) the locale, as is done with a [[formal topology]]. Once this structure is assumed, one can then ask whether a formal topology is overt, i.e. whether the axiomatic positivity predicate satisfies the requisite adjointness.



\end{itemize}
In impredicative constructive mathematics, overt locales can be characterized by the \emph{positive covering lemma}: $X$ is overt iff every open $U\in O(X)$ can be covered by positive opens, and iff every covering of an open $U$ can be refined to a covering by positive elements.

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

\begin{itemize}%
\item At least in impredicative constructive mathematics, the [[topological locale|locale induced by a topological space]] is overt. An open subspace $U$ is positive if and only if it is inhabited.


\item At least in impredicative constructive mathematics, the [[spectrum of a commutative ring]] $A$ (defined as the locale whose frame is the frame of radical ideals) is overt if and only if any element of $A$ is nilpotent or not nilpotent.



\end{itemize}
\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

As compact spaces go with [[proper maps]], so overt spaces go with [[open maps]]. Indeed, $X$ is compact if and only if the unique map $X \to pt$ to the [[point]] is proper, while $X$ is overt if and only if the unique map $X \to pt$ is open. Similarly, if $X\to pt$ is instead [[closed map|closed]], then $X$ is [[covert space|covert]].

Note that overtness is expressed only in terms of a [[left adjoint]], whereas open maps of locales must additionally satisfy a [[Frobenius reciprocity]] condition. In the case of locale maps to the point, this latter condition is automatic.

An overt space may also be called \textbf{locally $(-1)$-connected}, since this condition is the [[(0,1)-topos]]-theoretic version of the notion of [[locally connected topos]] and [[locally n-connected (n,1)-topos]]. A similar thing happens for higher local connectivity involving the Frobenius reciprocity condition, which must be imposed on general [[geometric morphisms]] to make them locally $n$-connected, but is automatic for morphisms to the point.

\hypertarget{in_synthetic_topology}{}\subsection*{{In synthetic topology}}\label{in_synthetic_topology}

In [[synthetic topology]], we interpret `space' to mean simply `set' (or [[type]], i.e. the basic objects of our foundational system). If the notion of `open' is specified by a [[dominance]], then there is an induced nontrivial notion of ``overt set'', defined essentially as above. For instance, the [[Rosolini dominance]] is the smallest dominance such that $\mathbb{N}$ is overt, whereas if all sets are overt then the dominance is trivial.

On the other hand, if by `open' we mean an open subset \href{/nlab/show/open+subspace#SynTop}{in the sense of Penon}, then all sets are overt.

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

The notion of an open [[locale]] was originally introduced by Joyal and Tierney (and developed by Johnstone in [[Stone Spaces]]):

\begin{itemize}%
\item [[Andre Joyal]], [[Myles Tierney]], \emph{An extension of the Galois theory of Grothendieck}, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71 pp.

\end{itemize}
The term ``overt'' is due to [[Paul Taylor]]. For example, it appears in:

\begin{itemize}%
\item [[Paul Taylor]], \emph{A lambda calculus for real analysis}, Journal of Logic and Analysis 2(5), pp.1-115, 2010. (\href{http://www.paultaylor.eu/ASD/lamcra/}{web})

\end{itemize}
Some of the history is described in the introduction to:

\begin{itemize}%
\item [[Bas Spitters]], \emph{Locatedness and Overt Sublocales}, Annals of Pure and Applied Logic, 162:1, October 2010, Pages 36--54. (\href{http://arxiv.org/abs/math/0703561}{arxiv})

\end{itemize}
[[!redirects overt space]] [[!redirects overt spaces]] [[!redirects open space]] [[!redirects open spaces]]

[[!redirects overt locale]] [[!redirects overt locales]] [[!redirects open locale]] [[!redirects open locales]]

[[!redirects locally (-1)-connected space]] [[!redirects locally (-1)-connected spaces]] [[!redirects locally (-1)-connected locale]] [[!redirects locally (-1)-connected locales]]

[[!redirects overt]] [[!redirects overtness]]



\end{document}