Interpreting Topological Logics over Euclidean Spaces

In this paper, we consider propositional Topological logics are a family of languages for representing topological logics with connectedness, i.e. topological and reasoning about topological data. The non-logical logics in which the only logical connectives are the usual primitives of these languages stand for various topological Boolean operators, but where there is a non-logical primitive relations and operations, and their valid formulas encode our expressing the property of topological connectedness knowledge about those relations and operations. Consider, (or a variant thereof). We show that such topological logics for example, the six relations illustrated in Figure 1. By em-are typically sensitive both to the spaces they are interpreted over and--more particularly--to the subsets of those spaces over which their variables are allowed to range.