Filling the gaps in an unpublished example of Nyikos: a countably compact non-compact manifold under $\clubsuit_C$

Abstract

We provide details for a Theorem which is only available on a unfinished preliminary draft of P. Nyikos: the existence under $\clubsuit_C$ of a hereditarily collectionwise normal countably compact non-compact manifold which does not contain a copy of $ω_1$. This shows in particular that {\bf MA + $\neg$CH} does not imply the existence of a copy of $ω_1$ in a countably compact non-compact manifold (it is known that {\bf 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 $ω_1$ and $\mathbb{Z}_n$-``bundles'' over $ω_1$, where $\mathbb{Z}_n$ is the additive group of integers modulo $n$. A small generalization of $\clubsuit_C$ is also quickly investigated.