# Franklin Tall: PFA(S)[S] and locally countable subspaces of compact countably tight spaces

Place: Fields Institute (Room 210)
Date: 19-June-2015 (13:30-15:00)
Speaker: Franklin Tall.
Title: PFA(S)[S] and locally countable subspaces of compact countably tight spaces.
Abstract: I have lectured many times in the seminar on Stevo’s method of forcing with a coherent Souslin tree S over a model of PFA restricted to posets that preserve S, since it has many interesting applications in set-theoretic topology. However I believe the current cohort of graduate students has not seen an actual proof of this sort. Since the seminar is suffering from a lack of speakers, I plan to give a sporadic series of lectures featuring such proofs. In particular, as soon as I understand it sufficiently well, I want to give Alan Dow’s proof that in such models, first countable perfect pre-images of omega_1 include copies of omega_1. This is the capstone of the proof of the consistency of every hereditarily normal manifold of dimension > 1 being metrizable. First of all, however, I want to prove a technical theorem that is necessary for the manifold result, and for many other results concerning under what conditions locally compact normal spaces are paracompact. This particular theorem – getting locally countable collections to be sigma-discrete – is perhaps not of wide interest, but the method of getting an uncountable set in such a model to be the union of countably many “nice” subsets (rather than just including an uncountable nice subset) should have more applications. The “proof” I gave of this result in the seminar five years ago turned out to have a gap. The gap is bridged by a clever idea of Stevo. The proof will appear in a joint paper.

The proof will be more digestible if listeners have had some exposure to proper forcing.