Regarding Overlapping as a Basic Concept of Subset Spaces

An operator describing overlapping is substituted/or the effort modality of Moss and Parikh´s logic of knowledge and topology. This means that the spatial idea underlying that system, viz closeness, is replaced with covering. We ask what properties of frames can be expressed by means of the new language. It turns out that the language is, for example, capable of characterizing directed spaces. After adding some expressive power originating from hybrid logic we are able to deal with linearity as well. This makes a temporal interpretaion of the new system possible. - The main issues of this paper are first corresponding completeness theorems, and second the decidability of the arising logics.