Kuznetsov’s Problem for spaces (topological completeness of si-logics)

Determine whether every superintuitionistic propositional logic (si-logic) is the logic of some class of topological spaces; equivalently, decide whether topologically incomplete si-logics exist.

Background

Kripke incompleteness for si-logics is known, but whether there exist si-logics that are not representable as logics of topological spaces remains unsettled. Resolving this would settle a long-standing problem dating back to Kuznetsov (1975).

The chapter frames this as a central open problem in the landscape of intuitionistic semantics, connecting Kripke, topological, and possibility-theoretic approaches.