Modal logic and topology

You can’t say much about modal logic in general. You have to be more specific to get anywhere. You have to choose some axioms. Ideally the axioms you need for your application correspond to a named set of axioms that has been studied before. The situation is similar in point-set topology. You can’t say very much about a general topological space. You have to specify some separation axioms to get going. Bare bones Modal logic A modal logic is any set of formulas in the modal language that: contains all propositional tautologies, is closed under modus ponens, and is closed under uniform substitution. In particular, this definition requires nothing of the modal operator □ (“box”). You just have propositional logic with a funny symbol added that could mean anything. Topology A topological space is a set X along with a set of subsets of X called open sets. The empty set and the full space X are open sets. Furthermore, the set of open sets is closed under finite intersections and arbitrary…