On modal logics arising from scattered locally compact Hausdorff spaces
Guram Bezhanishvili, Nick Bezhanishvili, Joel Lucero-Bryan, Jan van Mill
Abstract:
For a topological space X, let L(X) be the modal logic of X where \Box is interpreted as interior (and hence \Diamond as closure) in X. It is known that the modal logics S4, S4.1, S4.2, S4.1.2, S4.Grz, S4.Grz_n, 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 \geq 1. In fact, we prove this result for any scattered locally compact open hereditarily collectionwise normal and open hereditarily strongly zero-dimensional space.