无限紧Hausdorff空间$T$对应的环$C(T)$的谱中紧开子集的代数
The Algebra of compact-open subsets in the spectrum of the ring $C(T)$ for an infinite compact Hausdorff space $T$
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中文总结 AI 辅助
本研究给出交换Bezout环的紧开子集格成为Heyting代数的判别准则，通过构造反例彻底解决Bezhanishvili-Tressl问题，证明不存在无限紧Hausdorff空间$T$使$\text{Spec}(C(T))$为Esakia空间。
中文摘要 AI 辅助
对于交换Bezout环$R$，我们给出了$\text{Spec}(R)$的紧开子集格$\ring{\u25eb{K}}(\text{Spec}(R))$成为Heyting代数的判别准则，该准则基于具有主根的冒号理想。Bezhanishvili和Tressl证明了当$T$是基本不连通紧Hausdorff空间时，$\ring{\u25eb{K}}(\text{Spec}(C(T)))$是伪补的，并提出$\text{Spec}(C(β\bb{N}))$是否实际上是Esakia空间的问题。我们证明它不是：将我们的准则应用于$C(β\bb{N}) \u2245 \u2113^\u221e(\bb{N},\bb{R})$可得到一个对角反例，且相同的阻碍排除了所有无限离散集$D$对应的$βD$。借助$\u03c3$-完备Boolean代数的网格存在定理，我们将该构造推广到所有基本不连通紧Hausdorff空间，彻底解决了Bezhanishvili-Tressl问题：不存在无限紧Hausdorff空间$T$使得$\text{Spec}(C(T))$是Esakia空间。
英文摘要
For a commutative Bezout ring R, we give a criterion, in terms of colon ideals with principal radical, for the lattice $\mathring{\mathcal{K}}(\mathrm{Spec}(R))$ of compact open subsets of $\mathrm{Spec}(R)$ to be a Heyting algebra. Bezhanishvili and Tressl showed that $\mathring{\mathcal{K}}(\mathrm{Spec}(C(T)))$ is pseudocomplemented whenever T is a basically disconnected compact Hausdorff space, and asked whether $\mathrm{Spec}(C(β\mathbb{N}))$ is actually an Esakia space. We show it is not: applying our criterion to $C(β\mathbb{N}) \cong \ell^\infty(\mathbb{N},\mathbb{R})$ produces a diagonal counterexample, and the same obstruction rules out $βD$ for every infinite discrete D. A grid-existence theorem for $σ$-complete Boolean algebras lets us push the construction to every basically disconnected compact Hausdorff space, settling the Bezhanishvili-Tressl question completely: for no infinite compact Hausdorff space T is $\mathrm{Spec}(C(T))$ an Esakia space.
