Computer programing for construction and enumeration of all regular topologies and equivalence relations on finite sets

This paper characterizes the regular principal topology on a nonempty set X. This result gives us a correspondence between the families of all regular principal toplogies on X and all equivalence relations on X. The strictly weaker regular principal topologies on a set X than a given regular principal topology on X are characterized. If Xn is a finite set of n points this paper mainly describes an algorithm to construct and enumerate all weaker regular and all strictly weaker regular principal topologies on Xn than a given regular topology on Xn. All regular topologies on Xn are constructed and enumerated.