Investigation of the topological interpretation of modal logics

This thesis studies interpretations of modal logics in topological spaces. We prove that adding a new axiom (P!P) to S4 gives us a sound and complete axiom system over all discrete topological spaces. We show non-completeness of S4 over three families of topological spaces: particular point, excluded point, and another natural generalization of the Sierpi nskispaces. Namely, we nd extensions of S4 that are sound over these families, although their completeness remains an open question. We show that givenany set X and any interpretation of in X that satis esS4, the image of this interpretation is a topology onX. We also study the in uence of themodal axioms of S4 on topological properties of the image.