  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.
