It is shown to be consistent that countable, Fréchet,α 1-spaces are first countable. The result is obtained by using a countable support iteration of proper partial orders of lengthω 2.

We construct a model in which the splitting number is large and every ultrafilter has a small subset with no pseudo-intersection.

Partitioner algebras are defined by Baumgartner and Weese 619) as a natural tool for studying the properties of maximal almost disjoint families of subsets of ω. We prove from PFA+ and that there exists a partitioner algebra which contains a subalgebra which is not representable as a partitioner algebra.