Basic tools and continuity-like properties in relator spaces

This paper provides the unification of several continuity-like properties of functions and relations in the framework of relator spaces. Motivated by Galois connections, we consider an ordered pair of relations instead of a single relation. A family R of relations on a set X to another set Y is called a relator on X to Y . All reasonable generalizations of the usual topological structures (such as proximities, closures, topologies, filters and convergences, for instance) can be derived from relators. Therefore, they should not be studied separately. From the various topological and algebraic structures (such as lower bounds, minimum and infimum, for instance) derived from relators, by using Pataki connections, we can obtain several closure and projection operations for relators. Each of them will lead to four continuity-like properties of an ordered pair of relators.