Baire lattice

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.

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).

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.

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

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 $d \in N$.

Every continuous lattice is a Baire lattice.

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.

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, M. Mislove, Dana Scott,
*Continuous Lattices and Domains*2003, Vol. 93 of*Encyclopedia of Mathematics and its Applications*(doi:10.1017/CBO9780511542725)

Last revised on April 25, 2021 at 12:34:59. See the history of this page for a list of all contributions to it.