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arXiv · 2608.08251

The Algebra of compact-open subsets in the spectrum of the ring $C(T)$ for an infinite compact Hausdorff space $T$

Evgeny Kuznetsov
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Abstract

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.

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BibTeXRIS

Evgeny Kuznetsov. 2026-08-08. The Algebra of compact-open subsets in the spectrum of the ring $C(T)$ for an infinite compact Hausdorff space $T$. https://arxiv.org/abs/2608.08251

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