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By James E. Baumgartner


PFA(S)[S] and the Arhangel’skiĭ-Tall problem. (English summary) Topology Proc. 40 (2012), 99–108. The Arhangel ′ skii–Tall problem was posed by F. D. Tall [Canad. J. Math. 26 (1974), 1–6; MR0377823 (51 #13992)], who asked if the following statement is true: (AT) Every normal, locally compact, metacompact space is paracompact. The answer turned out to be independent of ZFC. P. B. Larson and Tall [Fund. Math. 210 (2010), no. 3, 285–300; MR2733053 (2011j:54016)] showed that, assuming the consistency of the existence of a supercompact cardinal, it is consistent that (LT) Every perfectly normal, locally compact space is paracompact. The relevant model was obtained by choosing a suitable ground universe with a coherent Souslin tree S, forcing (using PFA) restricted to proper forcings preserving S (i.e., PFA(S)) and then forcing with S. In the present article the author notices that the conjunction of the two statements (AT) and (LT) holds in the same model described above. This motivates him to further investigate models of the form PFA(S)[S], MAω1 (S)[S], etc., in the context of statements (AT) and (LT) and their relatives. Reviewed by Andrzej Rosłanowski References 1. U. Abraham and S. Todorčević, Martin’s axiom and first-countable S- and Lspaces. Kune

Year: 2013
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