nLab Baire lattice

Idea

Baire lattices are lattice-theoretic abstraction of the Baire property among topological spaces. 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.

Related pages

• complete lattice
• Baire category theorem
• Baire space

References

The concept appeared in:

• Karl Heinrich Hofmann, A note on Baire spaces and continuous lattices 1980. Bulletin of the Australian Mathematical Society, 21(2), pp. 265-279. (doi:10.1017/S0004972700006080)

Textbook accounts:

• G. Gierz, Karl Heinrich Hofmann, K. Keimel, J. D. Lawson, Michael Mislove, Dana Scott, Continuous Lattices and Domains 2003, Vol. 93 of Encyclopedia of Mathematics and its Applications (doi:10.1017/CBO9780511542725)