In the paper, we investigate (scattered) compact spaces with a $P$-base for some poset $P$. More specifically, we prove that any compact space with an ${\omega }^{\omega }$-base is metrizable and any scattered compact space with an ${\omega }^{\omega }$-base is countable under the assumption ${\omega }_1<\mathfrak{b}$. These give positive solutions to Problems 8.6.9 and 8.7.7 in Banakh (2019). Using forcing, we also prove that in a model of ${\omega }_1<\mathfrak{b}$, there is a non-first-countable compact space with a $P$-base for some poset $P$ with calibre ${\omega }_1$.

中文翻译：

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带 P 型底座的紧凑空间

在论文中，我们研究了（分散的）紧致空间 $磷$- 一些poset的基础 $磷$. 更具体地说，我们证明了任何具有${\omega }^{\omega }$-base 是可测量的，并且任何分散的紧凑空间具有 ${\omega }^{\omega }$-base 在假设下是可数的 ${\omega }_1<\mathfrak{乙}$. 这些为 Banakh (2019) 中的问题 8.6.9 和 8.7.7 提供了积极的解决方案。使用强迫，我们还证明了在一个模型中${\omega }_1<\mathfrak{乙}$，存在一个不可数的紧空间 $磷$- 一些poset的基础 $磷$ 带口径 ${\omega }_1$.