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

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 $\mathrm{Spec}(C(\beta \mathbb{N}))$ is actually an Esakia space. We show it is not: applying our criterion to $C(\beta \mathbb{N})\cong {\ell }^{\infty }(\mathbb{N},\mathbb{R})$ produces a diagonal counterexample, and the same obstruction rules out $\beta D$ for every infinite discrete D. A grid-existence theorem for $\sigma$-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.