Generalizing ordinary topological and pretopological spaces, we introduce the notion of peritopology where neighborhoods of a point need not contain that point, and some points might even have an empty neighborhood. We brie y describe various intrinsic aspects of this notion. Applied to modal logic, it gives rise to peritopological models, a generalization of topo- logical models, a spacial case of neighborhood semantics. A new cladding for bisimulation is presented. The concept of Alexandro peritopology is used in order to determine the logic of all peritopo- logical spaces, and we prove that the minimal logic K is strongly complete with respect to the class of all peritopological spaces. We also show that the classes of T0, T1 and T2-peritopological spaces are not modal denable, and that D is the logic of all proper peritopological spaces. Finally, among our conclusions, we show that the question whether T0, T1 peritopological spaces are modal denable in H(@) remains open.

Generalizing ordinary topological and pretopological spaces, we introduce the notion of peritopology where neighborhoods of a point need not contain that point, and some points might even have an empty neighborhood. We brie y describe various intrinsic aspects of this notion. Applied to modal logic, it gives rise to peritopological models, a generalization of topo- logical models, a spacial case of neighborhood semantics. A new cladding for bisimulation is presented. The concept of Alexandro peritopology is used in order to determine the logic of all peritopo- logical spaces, and we prove that the minimal logic K is strongly complete with respect to the class of all peritopological spaces. We also show that the classes of T0, T1 and T2-peritopological spaces are not modal denable, and that D is the logic of all proper peritopological spaces. Finally, among our conclusions, we show that the question whether T0, T1 peritopological spaces are modal denable in H(@) remains open.