Science and Education Publishing Journal of Mathematical Sciences and Applications 2333-8792 4 1 2016 10 29 The Relationship between the Topological Properties and Common Modal Logics 29 33 EN Maria Nogin Department of Mathematics, California State University, Fresno Bing Xu JMSA2016415 10.12691/jmsa-4-1-5 2016 6 12 2016 9 28 2016 10 27 A modal language is the language of the classical logic extended by additional operator(s), e.g. . Modal logics have a variety of interpretations and applications in different sciences, and depending on the context, different axioms involving may be assumed. In topological interpretations, the operator interpreted as interior. It is well known that the modal logic S4 is sound and complete over all topological spaces. In this paper we reverse the question. Given a set X and any interpretation of in X that satisfies a given subset of the axioms of S4, we determine which topological properties must be possessed by the image of the interpretation of .