nLab
Baire lattice

Contents

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,bLa, b \in L, we say that aa is way below bb and write aba \ll b if whenever SLS \subseteq L is a directed subset and bSb \leq \bigvee S (where S\bigvee S denotes the join of SS), then there exists sSs \in S with asa \leq s. Further we say that LL is continuous if for every aLa\in L, the subset

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

is directed and has join aa.

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 pLp \in L is called irreducible if ab=pa \vee b = p implies p=ap = a or p=bp = b.

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

Definition

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

Definition

A complete lattice LL is called a Baire lattice if for any countable family of dense elements NLN \subset L and each nonzero element uLu \in L there is an irreducible element pLp \in L such that dup d \wedge u \nleq p for all dNd \in N.

Theorem

Theorem

Every continuous lattice is a Baire lattice.

Example

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

References

The concept appeared in:

Textbook accounts:

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