PFA(S)[S] and Locally Compact Normal Spaces

关于正规第一可数线性H-闭空间的三个小结果

Three small results on normal first countable linearly H-closed spaces

本研究借助PFA、MA$_{ω_1}$(S)[S]与PFA(S)[S]的已知拓扑推论，证明在对应模型下，带多种附加性质的正规第一可数线性H-闭空间均为紧空间，补充了相关拓扑空间的紧性结论。

On the hereditary paracompactness of locally compact, hereditarily normal spaces

We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space which does not include a perfect pre-image of omega_1 is hereditarily paracompact.

Locally compact, $ω_1$-compact spaces

An $ω_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $ω_1$-compact space is $σ$-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties. Many results shown here are independent of the usual (ZFC) axioms of set theory, and the consistency of some may involve large cardinals. For example, it is independent of the ZFC axioms whether every locally compact, $ω_1$-compact space of cardinality $\aleph_1$ is $σ$-countably compac…

Notes on linearly H-closed spaces and od-selection principles

A space is called linearly H-closed iff any chain cover possesses a dense member. This property lies strictly between feeble compactness and H-closedness. While regular H-closed spaces are compact, there are linearly H-closed spaces which are even collectionwise normal and Fréchet-Urysohn. We give examples in other classes, and ask whether there is a first countable normal linearly H-closed non-compact space in ZFC. We show that PFA implies a negative answer if the space is moreover either locally separable or locally compact and locally ccc. Ostaszewski space (built with $\diamondsuit$) is an…

$γ^*$-Regular,$γ$-Locally Compact and $γ$-Normal spaces

We continue studying the properties of $γ_0$-compact, $γ^*$-regular and $γ$-normal spaces defined in [5]. We also define and discuss $γ$-locally compact spaces.

Normality versus paracompactness in locally compact spaces

This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on ω_1, as well as of a strong form of Chang's Conjecture. Together with other improvements, this enables the characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of ω_1.