Applications of some strong set-theoretic axioms to locally compact T5 and hereditarily scwH spaces

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 Lindelof number less than or equal to N-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 T5 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.