Generalized topological semantics for weak modal logics

Generalized topological spaces are not necessarily closed under finite intersections. Moreover, the whole universe does not need to be open. We use modified version of this framework to establish certain models for non-normal modal logics. We consider at least two approaches to this topic, wherein one of them is based on the interplay of two operators, both reflecting the idea of necessity. The second one leads to the quite known logics MT4 and MNT4. We obtain some completeness results and we prove correspondence between different classes of frames. Also, three types of bisimulation are investigated.