Applications of some strong set-theoretic axioms to locally compact T$_5$ and hereditarily scwH spaces

Peter J. Nyikos¹

¹ Department of Mathematics University of South Carolina Columbia, SC 29208, U.S.A.

Fundamenta Mathematicae, Tome 176 (2003) no. 1, pp. 25-45.

Résumé

Under some very strong set-theoretic hypotheses, hereditarily normal spaces (also referred to as T${}_5$ 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 $\omega _1$-compact spaces and another (Theorem 4) to all such spaces of Lindelöf number $\le \aleph _1$. 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 $\omega _1$. It also exposes (Theorem 2) the fine structure of perfect preimages of $\omega _1$ which are T${}_5$ and hereditarily collectionwise Hausdorff. In these theorems, “T${}_5$ 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.

Détail

DOI : 10.4064/fm176-1-3

Keywords: under strong set theoretic hypotheses hereditarily normal spaces referred spaces locally compact hereditarily collectionwise hausdorff have highly simplified structure paper gives structure theorem theorem applies omega compact spaces another theorem spaces lindel number aleph introduces axiom axiom crowding functions consequences theorem crowding countably compact subspaces certain continuous preimages omega exposes theorem fine structure perfect preimages omega which hereditarily collectionwise hausdorff these theorems hereditarily collectionwise hausdorff weakened hereditarily strongly collectionwise hausdorff corollaries include consistency modulo large cardinals every hereditarily strongly collectionwise hausdorff manifold dimension being metrizable concept alignment plays important role formulating several structure theorems