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\begin{document}

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\section*{Baire lattice}

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

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

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

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

A Baire lattice is a [[lattice|lattice theoretic]] abstraction of the [[Baire space|Baire property]] of a topological space. In parallel to the fact that every [[complete space|complete metric space]] is a Baire space every [[continuous poset|continuous lattice]] is a Baire lattice.

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

Recall that a [[complete lattice]] is a [[partial order|poset]] which has all small [[join|joins]] and [[meet|meets]]. For $a, b \in L$, we say that $a$ is \textbf{way below} $b$ and write $a \ll b$ if whenever $S \subseteq L$ is a [[directed subset]] and $b \leq \bigvee S$ (where $\bigvee S$ denotes the [[join]] of $S$), then there exists $s \in S$ with $a \leq s$. Further we say that $L$ is \textbf{[[continuous lattice|continuous]]} if for every $a\in L$, the subset

\begin{displaymath}
\Downarrow (a) \coloneqq \{ b \in L | b \ll a \}
\end{displaymath}
is directed and has join $a$.

For example the [[category of open subsets|lattice of open subsets]] of a topological space is a continuous lattice if and only if the [[sobrification]] of the topological space is [[locally compact]] (i.e. the topology has a basis of compact neighborhoods).

$\backslash$begin\{defn\} An element $p \in L$ is called \textbf{irreducible} if $a \vee b = p$ implies $p = a$ or $p = b$. $\backslash$end\{defn\}

To illustrate this definition think of an [[irreducible topological space|irreducible subset]] of a topological space.

$\backslash$begin\{defn\} An element $d\in L$ is \textbf{dense} if for all $a\in L$ the relation $a \neq \bot$ implies that $d \wedge a \neq \bot$. $\backslash$end\{defn\}

$\backslash$begin\{defn\} A [[complete lattice]] $L$ is called a \textbf{Baire lattice} if for any countable family of dense elements $N \subset L$ and each nonzero element $u \in L$ there is an irreducible element $p \in L$ such that $d \wedge u \nleq p$ for all $d \in N$. $\backslash$end\{defn\}

\hypertarget{theorem}{}\subsection*{{Theorem}}\label{theorem}

$\backslash$begin\{theorem\} Every continuous lattice is a Baire lattice. $\backslash$end\{theorem\}

\hypertarget{example}{}\subsection*{{Example}}\label{example}

Let $X$ be a [[sober space|sober]] [[topological space]]. Then the lattice of opens of $X$ is Baire if and only if the topological space $X$ has the [[Baire space|Baire property]].

\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item [[complete lattice]]
\item [[Baire category theorem]]
\item [[Baire space]]

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

The concept appeared in

\begin{itemize}%
\item Karl H. Hofmann, \emph{A note on Baire spaces and continuous lattices} 1980. Bulletin of the Australian Mathematical Society, 21(2), pp. 265-279.

\end{itemize}
For a textbook reference see

\begin{itemize}%
\item G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, [[Dana Scott|D. S. Scott]], \emph{Continuous Lattices and Domains} 2003, Vol. 93 of \emph{Encyclopedia of Mathematics and its Applications}.

\end{itemize}


\end{document}