  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.
