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\begin{document}

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\section*{dense subspace}

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

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

[[!include topology - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{in_constructive_mathematics}{In constructive mathematics}\dotfill \pageref*{in_constructive_mathematics} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Given a [[topological space]] (or [[locale]]) $X$, a [[subspace]] $A$ of $X$ is \textbf{dense} if its [[closed subspace|closure]] is all of $X$: $cl(A)=X$.

Since $cl(A)$ is the set of all points $x$ such that every [[open neighborhood]] of $x$ [[intersection|intersects]] $A$, this can equivalently be written as ``every open neighborhood of every point intersects $A$'', or equivalently ``every [[inhabited set|inhabited]] open set intersects $A$'', i.e. $A\cap U$ is [[inhabited]] for all inhabited open sets $U$.

Contraposing this, we obtain another equivalent definition ``the only open subset not intersecting $A$ is the empty set'', or ``if $A\cap U=\emptyset$ for some open set $U$, then $U=\emptyset$''. This is the definition usually given when $X$ is a [[locale]]: a [[nucleus]] $j$ is dense if $j(0)=0$ (since $j(0)$ is the union of all opens whose ``intersection with $j$'' is $0$).

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

\begin{itemize}%
\item If $A \subseteq X$ is a dense subset of a topological space $X$ and $f: X \to Y$ is an [[epimorphism]], then the [[image]] $f(A)$ is dense in $Y$.


\item If i: $A \hookrightarrow X$ and $j: X \hookrightarrow B$ are dense subspace inclusions, then so is the composite $j \circ i: A \to B$.


\item If $A\subseteq X$ is a dense subset of topological space $X$ and $A$ is connected, so is $X$.


\item In point-set topology, a space is [[separable space|separable]] if and only if it has a dense subspace with [[countable set|countably]] many points.


\item In [[locale]] theory, we have the curious property that any [[intersection]] of dense subspaces is still dense. (This of course fails rather badly for [[topological spaces]], where the intersection of all dense topological subspaces is the space of [[isolated point]]s.) One consequence is that every locale has a smallest dense sublocale, the [[double negation sublocale]].



\end{itemize}
\hypertarget{in_constructive_mathematics}{}\subsection*{{In constructive mathematics}}\label{in_constructive_mathematics}

In [[constructive mathematics]], the law of contraposition is not an equivalence, so we obtain two inequivalent notions of density:

\begin{itemize}%
\item $A\subseteq X$ is \textbf{strongly dense} if $A\cap U$ is inhabited for all inhabited open sets $U\subseteq X$.
\item $A\subseteq X$ is \textbf{weakly dense} if $A\cap U = \emptyset$ for some open $U\subseteq X$ implies $U=\emptyset$.

\end{itemize}
Of course, strong density implies weak density, since emptiness is non-inhabitation (whereas inhabitation is stronger than non-emptiness). The two notions of density are related dually to the corresponding notions of [[closed subspace]]: $A$ is strongly dense iff its weak closure is all of $X$, and weakly dense iff its strong closure is all of $X$.

Note that the usual notion of density for [[sublocales]] $j(0)=0$ is an analogue of \emph{weak} density, and could be called such. There is also a notion of \emph{strong} density for sublocales. Since strong density refers to inhabited sets, one might expect strong density for sublocales to refer to [[positive elements]], and thus only be sensible for [[overt locales]]; but in fact it can be reformulated to make sense in all cases.

\begin{udefn}
A [[nucleus]] $j$ on a locale $X$ is \textbf{strongly dense} if $j(\hat{P})=\hat{P}$ for any [[truth value]] $P$, where $\hat{P} = \bigvee \{ X \mid P \}$.

\end{udefn}
With classical logic, every truth value is either $\top$ or $\bot$, and we have $\hat{\top}=X$ (and any nucleus satisfies $j(X)=X$) while $\hat{\bot}=0$. Thus classically strong and weak density coincide. To see that this is really a notion of strong density, we prove:

\begin{utheorem}
If $i:A\subseteq X$ is a sublocale such that $A$ and $X$ are both [[overt]], then $A$ is strongly dense if and only if for any positive open $U\in O(X)$, the intersection $A\cap U = i^*U \in O(A)$ is also positive.

\end{utheorem}
\begin{proof}
First suppose $A$ is strongly dense, and let $U\in O(X)$ be positive. Let $P$ be the truth value of the statement ``$i^*U \in O(A)$ is positive''. We want to show that $P$ is true, for which it suffices to show that $\hat{P} = \bigvee \{ X \mid P \}$ is positive, since then its covering $\{ X \mid P \}$ would be inhabited and thus $P$ would be true. And since $U$ is positive, it suffices to show $U \subseteq \hat{P}$.

Now since $A$ is strongly dense, $j_A(\hat{P}) = \hat{P}$, which is to say that $\hat{P} = i_*(i^*\hat{P})$. By adjointness, therefore, to show $U \subseteq \hat{P}$ it suffices to show $i^*U \subseteq i^*\hat{P} = \bigvee \{ A \mid P\}$. Now since $A$ is overt, $i^*U$ can be covered by positive opens, so it suffices to show that for any positive $V\subseteq i^*U$ we have $V\subseteq \bigvee \{ A \mid P\}$. But if $V\subseteq i^*U$ is positive, then $i^*U$ is also positive, i.e. $P$ is true, and thus $\bigvee \{ A \mid P\} = A$, which contains $V$.

Now suppose conversely that for any positive $U\in O(X)$, $i^*U$ is also positive, and let $P$ be any truth value; we must show $i_* i^*\hat{P} \subseteq \hat{P}$. Since $X$ is overt, $i_* i^*\hat{P}$ can be covered by positive opens, so it suffices to show that for any positive $U\subseteq i_* i^*\hat{P}$ we have $U\subseteq \hat{P}$. But by adjointness, $U\subseteq i_* i^*\hat{P}$ is equivalent to $i^*U \subseteq i^*\hat{P}$, and by assumption $i^*U$ is also positive. Thus, $i^*\hat{P} = \bigvee \{ A \mid P\}$ is positive, which means that $P$ is true, and hence $\hat{P} = X$ and so $U\subseteq \hat{P}$.

\end{proof}
Since spatial locales are overt, and their positivity predicate coincides with inhabitedness, we have in particular:

\begin{ucor}
If $i:A\subseteq X$ is a subspace of a topological space, then $A$ is strongly dense as a topological subspace if and only if it is strongly dense as a sublocale.

\end{ucor}
Strong density for sublocales gives rise to a corresponding notion of [[weakly closed sublocale]]. It is also the specialization of the notion of [[fiberwise dense sublocale]] to the case of locale maps $X\to 1$.

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

\begin{itemize}%
\item [[dense subalgebra]]


\item [[dense subtopos]]


\item [[closed subspace]]


\item [[nowhere dense set]]



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

Strongly dense sublocales are discussed in

\begin{itemize}%
\item [[Sketches of an Elephant]], C1.1 and C1.2


\item [[Peter Johnstone]], \emph{A constructive `closed subgroup theorem' for localic groups and groupoids}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques (1989), Volume: 30, Issue: 1, page 3-23 \href{https://eudml.org/doc/91430}{link}


\item M. Jibladze and [[Peter Johnstone]], \emph{The frame of fibrewise closed nuclei}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques (1991), Volume: 32, Issue: 2, page 99-112, \href{https://eudml.org/doc/91478}{link}


\item [[Peter Johnstone]], \emph{Fiberwise separation axioms for locales}



\end{itemize}
[[!redirects dense subspace]] [[!redirects dense subspaces]] [[!redirects dense subset]] [[!redirects dense subsets]] [[!redirects dense set]] [[!redirects dense sets]] [[!redirects dense topological subspace]] [[!redirects dense topological subspaces]]

[[!redirects strongly dense subspace]] [[!redirects strongly dense subspaces]] [[!redirects strongly dense subset]] [[!redirects strongly dense subsets]] [[!redirects strongly dense set]] [[!redirects strongly dense sets]] [[!redirects strongly dense topological subspace]] [[!redirects strongly dense topological subspaces]]

[[!redirects weakly dense subspace]] [[!redirects weakly dense subspaces]] [[!redirects weakly dense subset]] [[!redirects weakly dense subsets]] [[!redirects weakly dense set]] [[!redirects weakly dense sets]] [[!redirects weakly dense topological subspace]] [[!redirects weakly dense topological subspaces]]

[[!redirects dense sublocale]] [[!redirects dense sublocales]] [[!redirects dense nucleus]] [[!redirects dense nuclei]]

[[!redirects strongly dense sublocale]] [[!redirects strongly dense sublocales]] [[!redirects strongly dense nucleus]] [[!redirects strongly dense nuclei]]

[[!redirects weakly dense sublocale]] [[!redirects weakly dense sublocales]] [[!redirects weakly dense nucleus]] [[!redirects weakly dense nuclei]]



\end{document}