PFA(S)[S] and countably compact spaces

Abstract

We show a number of undecidable assertions concerning countably compact spaces hold under PFA(S) [S]. We also show the consistency without large cardinals of every locally compact, perfectly normal space is paracompact.

PFA(S)[S] for the masses

“…CTPPI was obtained by Alan Dow via a more complicated version of the proof for PPI + . This appears in [5].…”

“…The problem of whether it was consistent there are no counterexamples was raised by S. Watson [31], [32]. Larson and Tall [16] constructed the required model of PFA(S)[S]; A. Dow and Tall managed to drop the large cardinal [5].…”

“…Consistent counterexamples had earlier been constructed by G. Gruenhage 6. PFA(S)[S] implies countably compact, perfectly normal spaces are compact [5].…”

Normality Versus Paracompactness in Locally Compact Spaces

“…Theorem 2.10 [11]. PFA(S)[S] implies a countably tight, perfect pre-image of ω 1 includes a copy of ω 1 .…”

“…At least 45 years ago, it was recognized that subparacompactness plus collectionwise Hausdorffness would with "normal implies collectionwise Hausdorff" consequences of V = L was exploited in [26] in order to prove the consistency, modulo a supercompact cardinal, of every locally compact perfectly normal space is paracompact. The large cardinal was later removed, so that: Theorem 1.4 [11]. If ZFC is consistent, then so is ZFC plus every locally compact perfectly normal space is paracompact.…”