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 omega_1 is hereditarily paracompact.

💡 Research Summary

The paper investigates a long‑standing problem in topology: under what set‑theoretic hypotheses does every locally compact, hereditarily normal space become hereditarily paracompact? The authors prove a consistency result: assuming the existence of a supercompact cardinal, it is consistent that every locally compact, hereditarily normal space which does not contain a perfect pre‑image of ω₁ is hereditarily paracompact. The proof builds on a series of earlier papers by Larson and Tall, in which a particular kind of Souslin tree S is used as a “seed” for forcing constructions. The main technical steps are as follows.

1. Axioms and forcing notions. The authors work with Baumgartner’s proper forcing axiom PFA and its stronger variants PFA⁺ and PFA⁺⁺ (the latter called PFA⁺⁺ in the paper). They also use Axiom R (Fleissner’s reflection principle) and its stronger form R⁺⁺. The axiom Σ⁺⁺ and the combinatorial principle Σ′ are introduced to control countable tightness and discrete families in compact spaces.

2. **The model PF A⁺⁺(S)