Filling the gaps in an unpublished example of Nyikos: a countably compact non-compact manifold under ${♣}_C$

Abstract

We provide details for a Theorem which is only available on a unfinished preliminary draft of P. Nyikos: the existence under ${♣}_C$ of a hereditarily collectionwise normal countably compact non-compact manifold which does not contain a copy of ${\omega }_1$. This shows in particular that MA + $\neg$CH does not imply the existence of a copy of ${\omega }_1$ in a countably compact non-compact manifold (it is known that PFA does imply it). The said manifold is obtained from a principal ${\mathbb{S}}^1$-bundle over the long ray, and some structural theorems for these spaces (due to Nyikos as well) are also proved. We show that the same type of theorems hold for $n$-to-$1$ closed preimages of ${\omega }_1$ and ${\mathbb{Z}}_n$-``bundles'' over ${\omega }_1$, where ${\mathbb{Z}}_n$ is the additive group of integers modulo $n$. A small generalization of ${♣}_C$ is also quickly investigated.

Based on a preliminary draft by Peter Nyikos. V2: Added Subsection 6.6 and Lemma 2.21. Typos corrected