There has been a fundamental problem to determine the number of di?erent (non homeomorphic) topologies de?ned on a ?nite set of n points. This number has been determined for small numbers by enumeration, however, the general problem is very di?cult and has been so far remained open. (This number has been estimated asymptotically, indeed, it is known that the number of topologies on n points is asymptotically the same as the number of topologies on n points for which asymptotic bounds exists). The general problem of determination of the number of topologies on n points may be reduced to the problem of ?nding the number of topologies on n points with k open sets. We denote this number by T(n,k). As for the case of the general problem, this problem is also open, though several partial results exist. Our study in this thesis is divided into two parts.