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

### Volume 176 / 2003

#### Abstract

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.