Computer Construction and Enumeration of All T0 and All Hyperconnected T0 Topologies on Finite Sets

There are many axioms on the principal topological spaces. Two of the interesting axioms are the T0 and hyperconnected topological spaces. There is a well-known and straightforward correspondence (cf. [2]) between the topologies on finite set Xn of n points and reflexive transitive relations (preorders) on those sets. This paper generalizes this result, characterizes the principal hyperconnected T0-topologies on a nonempty set X and gives their number on a set Xn. It mainly describes algorithms for construction and enumeration of all weaker and strictly weaker T0 and nT0-topologieson on Xn. The algorithms are written in fortran 77 and implemented on pentium II400 system.