Halo Semantics for Modal Logic

Abstract

In nonstandard analysis the halo of a point in a topological space is the intersection of the nonstandard extensions of all its open neighbourhoods. We define a parametric family of modal operators from the halo by varying which elements of the nonstandard extension are admitted as witnesses, and identify four canonical instances. Two recover well-known modalities: the topological closure and the Cantor derivative. A third reduces to Kripke semantics over the specialisation preorder. The fourth, purely nonstandard instance admits only nonstandard witnesses. The Transfer Principle forces it to coincide with the $\omega$-accumulation point operator, a classical topological notion not previously studied in modal logic. Unlike the Cantor derivative, the $\omega$-accumulation operator maps arbitrary sets to closed sets without any separation axiom, yielding an $\omega$-Cantor-Bendixson decomposition on all topological spaces. Axiom 4 holds universally, again without separation conditions. We prove that K4 is the complete logic over infinite spaces, and GL over infinite $\omega$-scattered spaces.