It is consistent that there is a compact space of character less than the splitting number in which there are no converging sequences. Such a space is an Efimov space.
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.
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.
It was proved recently that Telgársky’s conjecture, which concerns partial information strategies in the Banach–Mazur game, fails in models of \. The proof introduces a combinatorial principle that is shown to follow from \, namely: \::Every separative poset \ with the \-cc contains a dense sub-poset \ such that \ for every \. We prove this principle is independent of \ and \, in the sense that \ does not imply \, and \ does not imply \ assuming the consistency (...) of a huge cardinal. We also consider the more specific question of whether \ holds with \ equal to the weight-\ measure algebra. We prove, again assuming the consistency of a huge cardinal, that the answer to this question is independent of \. (shrink)
