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# Contents

## Idea

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

## Definitions

Recall that a complete lattice is a poset which has all small joins and meets. For $a, b \in L$, we say that $a$ is 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 continuous if for every $a\in L$, the subset

$\Downarrow (a) \coloneqq \{ b \in L | b \ll a \}$

is directed and has join $a$.

For example the 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).

###### Definition

An element $p \in L$ is called irreducible if $a \vee b = p$ implies $p = a$ or $p = b$.

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

###### Definition

An element $d\in L$ is dense if for all $a\in L$ the relation $a \neq \bot$ implies that $d \wedge a \neq \bot$.

###### Definition

A complete lattice $L$ is called a 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 $d \in N$.

## Theorem

###### Theorem

Every continuous lattice is a Baire lattice.

## Example

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

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