Under some very strong set-theoretic hypotheses, hereditarily normal spaces (also referred to as T₅ spaces) that are locally compact and hereditarily collectionwise Hausdorff can have a highly simplified structure. This paper gives a structure theorem (Theorem 1) that applies to all such ω₁-compact spaces and another (Theorem 4) to all such spaces of Lindelöf number ≤ ℵ₁. It also introduces an axiom (Axiom F) on crowding of functions, with consequences (Theorem 3) for the crowding of countably compact subspaces in certain continuous preimages of ω₁. It also exposes (Theorem 2) the fine structure of perfect preimages of ω₁ which are T₅ and hereditarily collectionwise Hausdorff. In these theorems, "T₅ and hereditarily collectionwise Hausdorff" is weakened to "hereditarily strongly collectionwise Hausdorff." Corollaries include the consistency, modulo large cardinals, of every hereditarily strongly collectionwise Hausdorff manifold of dimension > 1 being metrizable. The concept of an alignment plays an important role in formulating several of the structure theorems.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm176-1-3,
author = {Peter J. Nyikos},
title = {Applications of some strong set-theoretic axioms to locally compact T5 and hereditarily scwH spaces},
journal = {Fundamenta Mathematicae},
volume = {177},
year = {2003},
pages = {25-45},
zbl = {1027.54008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm176-1-3}
}
Peter J. Nyikos. Applications of some strong set-theoretic axioms to locally compact T₅ and hereditarily scwH spaces. Fundamenta Mathematicae, Tome 177 (2003) pp. 25-45. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm176-1-3/