The Topology of Full and Weak Belief
Alexandru Baltag, Nick Bezhanishvili, Aybüke Özgün, Sonja Smets

Abstract:
We introduce a new topological semantics for belief logics in which the belief modality is interpreted as the interior of the closure of the interior operator. We show that the system wKD45, a weakened version of KD45, is sound and complete w.r.t. the class of all topological spaces. Moreover, we point out a problem regarding updates on extremely disconnected spaces that appears in the setting of Baltag et al. (2013) and show that our proposal for topological belief semantics on all topological spaces constitutes a solution for it. While generalizing the topological belief semantics proposed in Baltag et al. (2013) to all spaces, we model conditional beliefs and updates and give complete axiomatizations of the corresponding logics.

Keywords: Topological models;  Epistemic and doxastic logic;  Updates;  Conditional beliefs;  (Hereditarily) extremally disconnected spaces.