Halo Semantics for Modal Logic

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 ω-accumulation point operator, a classical topological notion not previously studied in modal logic.Unlike the Cantor derivative, the ω-accumulation operator maps arbitrary sets to closed sets without any separation axiom, yielding an ω-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 ω-scattered spaces.