Forcing with a coherent Souslin tree and locally countable subspaces of countably tight compact spaces

On the Hereditary Paracompactness of Locally Compact, Hereditarily Normal Spaces

“…The results established here were actually proved around 2004, modulo results of Todorcevic announced in 2002 (which now appear in [7] and [19]) and of the second author [16]. We delayed submission until a correct version of [16] existed.…”

“…Our plan is to find a model in which these two consequences and Axiom R hold, as well as normality implying (strongly) ℵ 1 -collectionwise Hausdorffness for the spaces under consideration. The model we will consider is of the same genre as those in [12], [19], [7], [11], and [16]. One starts with a particular kind of Souslin tree S, a coherent one, which is obtainable from ♦ or by adding a Cohen real.…”

“…This is the sixth in a series of papers ( [12], [19], [7], [11], [16] 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 do not 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 ω 1 ) was proved in [11], using essentially that locally compact perfectly normal spaces are locally hereditarily Lindelöf and first countable. Here we avoid these last two properties by combining the methods of [2] and [16].…”

Normality Versus Paracompactness in Locally Compact Spaces

“…A modification of such a proof produces a model in which the following proposition (see [15]) holds:…”

PFA(S)[S] and countably compact spaces

“…The only new point to observe is that each stage of the iteration producing − for compact sequential spaces is proper and S-preserving, and has cardinality ℵ 1 , by CH. "Proper and S-preserving" was shown in [8]. As in the PPI without large cardinals proof in [5], we can code up the necessary information about locally countable sets of size ℵ 1 so as to get a poset of size ℵ 1 for establishing an instance of − .…”