在无选择公理情况下通过超滤子刻画 $\mathbb{N}$-紧性与实紧性
Characterizations of $\mathbb{N}$-compactness and realcompactness via ultrafilters in the absence of the axiom of choice
AI总结：
本文在 ZF 集合论框架下考察 Herrlich-Chew 定理与 Hewitt 定理，证明前者无需选择公理，后者在可数多选公理下成立，并给出其在 ZF 中有效的修正形式及应用。
摘要：
本文涉及 Herrlich-Chew 定理，该定理断言：一个 Hausdorff 零维空间是 $\mathbb{N}$-紧的，当且仅当该空间中每个具有可数交性质的闭开超滤子都是固定的。本文还涉及 Hewitt 定理，该定理断言：一个 Tychonoff 空间是实紧的，当且仅当该空间中每个具有可数交性质的 $z$-超滤子都是固定的。这些定理的原始证明都使用了选择公理。本文的目的是证明 Herrlich-Chew 定理在 $\mathbf{ZF}$ 中成立，而 Hewitt 定理是否可能在某个 $\mathbf{ZF}$ 模型中为假仍是一个开放问题。文中证明，在每个满足可数多选公理的 $\mathbf{ZF}$ 模型中，Hewitt 定理都成立。本文给出了 Hewitt 定理的一个修正形式，并证明其在 $\mathbf{ZF}$ 中为真。还展示了所得结果的若干应用。
英文摘要：
This article concerns the Herrlich-Chew theorem stating that a Hausdorff zero-dimensional space is $\mathbb{N}$-compact if and only if every clopen ultrafilter with the countable intersection property in this space is fixed. It also concerns Hewitt's theorem stating that a Tychonoff space is realcompact if and only if every $z$-ultrafilter with the countable intersection property in this space is fixed. The axiom of choice was involved in the original proofs of these theorems. The aim of this article is to show that the Herrlich-Chew theorem is valid in $\mathbf{ZF}$, but it is an open problem if Hewitt's theorem can be false in a model of $\mathbf{ZF}$. It is proved that Hewitt's theorem is true in every model of $\mathbf{ZF}$ in which the countable axiom of multiple choice is satisfied. A modification of Hewitt's theorem is given and proved true in $\mathbf{ZF}$. Several applications of the results obtained are shown.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
