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.
Recall that a complete lattice is a poset which has all small joins and meets. For , we say that is way below and write if whenever is a directed subset and (where denotes the join of ), then there exists with . Further we say that is continuous if for every , the subset
is directed and has join .
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 is called irreducible if implies or .
To illustrate this definition think of an irreducible subset of a topological space.
An element is dense if for all the relation implies that .
A complete lattice is called a Baire lattice if for any countable family of dense elements and each nonzero element there is an irreducible element such that for all .
Every continuous lattice is a Baire lattice.
Let be a sober topological space. Then the lattice of opens of is Baire if and only if the topological space has the Baire property.
The concept appeared in:
Textbook accounts:
Last revised on January 20, 2024 at 11:11:05. See the history of this page for a list of all contributions to it.