More on d-Logics of Subspaces of the Rational Numbers

Abstract

We prove that each countable rooted K4-frame is a d-morphic image of a subspace of the space $\mathbb{Q}$ of rational numbers. From this we derive that each modal logic over K4 axiomatizable by variable-free formulas is the d-logic of a subspace of $\mathbb{Q}$. It follows that subspaces of $\mathbb{Q}$ give rise to continuum many d-logics over K4, continuum many of which are neither finitely axiomatizable nor decidable. In addition, we exhibit several families of modal logics finitely axiomatizable by variable-free formulas over K4 that d-define interesting classes of topological spaces. Each of these logics has the finite model property and is decidable. Finally, we introduce quasi-scattered and semi-scattered spaces as generalizations of scattered spaces, develop their basic properties, axiomatize their corresponding modal logics, and show that they also arise as the d-logics of some subspaces of $\mathbb{Q}$.