On the Hereditary Paracompactness of Locally Compact, Hereditarily Normal Spaces

We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space which does not include a perfect pre-image of ω1 is hereditarily paracompact. This is the fifth in a series of papers ([LTo], [L2], [FTT], [LT], [T1] being the logically previous ones) that establish powerful topological consequences in models of set theory obtained by starting with a particular kind of Souslin tree S, iterating partial orders that don’t destroy S, and then forcing with S. The particular case of the theorem stated in the abstract when X is perfectly normal (and hence has no perfect pre-image of ω1) was proved in [LT], using essentially that locally compact perfectly normal spaces are locally hereditarily Lindelöf and first countable. Here we avoid these two last properties by combining the methods of [B2] and [T1]. To apply [B2], we establish the new set-theoretic result that PFA (S)[S] implies Fleissner’s “Axiom R”. This notation is explained below; the model is a strengthening of those used in the previous four papers. AMS Subj. Class. (2010): Primary 54D35, 54D15, 54D20, 54D45, 03E65; Secondary 03E35.