AI 中文总结
研究在\(\clubsuit_C\)下可数紧致非紧致流形的相关性质，通过长射线上的主\(\mathbb{S}^1\)丛构造流形，证明相关结构定理，表明{\bf MA + \(\neg\)CH}不蕴含该流形中\(\omega_1\)副本的存在，并研究了相关推广。
AI 中文摘要
我们给出了一个仅存在于 P. Nyikos 未完成初稿中的定理的细节：在\(\clubsuit_C\)下存在一个遗传集态正规的可数紧致非紧致流形，它不包含\(\omega_1\)的副本。这特别表明，{\bf MA + \(\neg\)CH}并不意味着在可数紧致非紧致流形中存在\(\omega_1\)的副本（已知{\bf PFA}意味着存在）。所述流形是从长射线上的一个主\(\mathbb{S}^1\)丛得到的，并且还证明了这些空间的一些结构定理（同样归功于 Nyikos）。我们表明，对于\(\omega_1\)的\(n\)对\(1\)闭原像和\(\omega_1\)上的\(\mathbb{Z}_n\) - “丛”，同样类型的定理成立，其中\(\mathbb{Z}_n\)是模\(n\)的整数加法群。还快速研究了\(\clubsuit_C\)的一个小推广。
英文摘要
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 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 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.
CommentsBased on a preliminary draft by Peter Nyikos. V2: Added Subsection 6.6 and Lemma 2.21. Typos corrected
