Ultrafilter selection and Corson compacta

TL;DR

This paper investigates the ultrafilter selection property in Boolean algebras, linking it to Corson compactness and exploring topological and algebraic structures to understand its implications.

Contribution

It introduces the ultrafilter selection property, characterizes it for certain cardinalities, and examines its relation to Corson compactness and various algebraic structures.

Findings

Ultrafilter selection property is equivalent to non-Corson compactness for cardinality aleph-one.

Results on Lindelöf number in the pointwise topology related to ultrafilter selection.

Analysis of poset Boolean algebras, interval algebras, and semilattices in this context.

Abstract

We study the question which Boolean algebras have the property that for every generating set there is an ultrafilter selecting maximal number of its elements. We call it the ultrafilter selection property. For cardinality aleph-one the property is equivalent to the fact that the space of ultrafilters is not Corson compact. We also consider the pointwise topology on a Boolean algebra, proving a result on the Lindel\"of number in the context of the ultrafilter selection property. Finally, we discuss poset Boolean algebras, interval algebras, and semilattices in the context of ultrafilter selection properties.