On modal logics arising from scattered locally compact Hausdorff spaces

Abstract

On modal logics arising from scattered locally compact Hausdorff spaces Bezhanishvili, G.; Bezhanishvili, N.; Lucero-Bryan, J.; van Mill, J.Abstract. For a topological space X, let L(X) be the modal logic of X where is interpreted as interior (and hence ♦ as closure) in X. It was shown in [6] that the modal logics S4, S4.1, S4.2, S4.1.2, S4.Grz, S4.Grz n (n ≥ 1), and their intersections arise as L(X) for some Stone space X. We give an example of a scattered Stone space whose logic is not such an intersection. This gives an affirmative answer to [6, Question 6.2]. On the other hand, we show that a scattered Stone space that is in addition hereditarily paracompact does not give rise to a new logic; namely we show that the logic of such a space is either S4.Grz or S4.Grz n for some n ≥ 1. In fact, we prove this result for any scattered locally compact open hereditarily collectionwise normal and open hereditarily strongly zero-dimensional space.2010 Mathematics Subject Classification. 03B45; 54G12.

Tree-like constructions in topology and modal logic

“…For a scattered Hausdorff space, [5, Thm. 4.9] demonstrates that finite Cantor-Bendixson rank is characterized by the concept of modal Krull dimension-a topological analogue of the depth of an S4-frame-introduced in [4]. For a topological analogue of cluster size, we recall that a space X is resolvable provided it contains a dense subset whose complement is also dense.…”

“…Proof. Suppose G is an interior image of Y , say via g : Y → W \ C. Since X \ Y is nresolvable, by [4,Lem. 5.9], there is an onto interior mapping h : X \ Y → C. We extend g to f : X → W by setting f (x) = h(x) for x ∈ X \ Y .…”