eng Science and Education Publishing Journal of Mathematical Sciences and Applications 2333-8792 2016-10-29 4 1 29 33 10.12691/jmsa-4-1-5 JMSA2016415 article Maria Nogin mnogin@csufresno.edu 1 Bing Xu 1 Department of Mathematics, California State University, Fresno 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 . http://pubs.sciepub.com/jmsa/4/1/5/jmsa-4-1-5.pdf modal logic topological space