Choice-free Stone-style duality for Heyting algebras (analogue of UV-spaces)

Develop a choice-free, possibility-based topological duality for arbitrary Heyting algebras by identifying the most natural analogue of upper Vietoris spaces (UV-spaces) for Boolean algebras; formally construct the corresponding dual category and prove a duality theorem that represents every Heyting algebra without invoking choice principles.

Background

UV-spaces provide a choice-free topological duality for Boolean algebras in possibility semantics. An analogous structure for Heyting algebras would extend possibility semantics beyond complete Heyting algebras and connect to locales and nuclei-based representations.

The chapter explicitly leaves open the identification of such a structure and the associated duality, positioning it as a cornerstone for a fully general possibility semantics for intuitionistic logic.