On the cardinalities of finite topologies

Following the ideas of Sharp [2,3], we will give a partial answer to the question: “Let k be an integer, k ? 2. What is the smallest integer m for which there is a topology on m points with k open sets.” We state several results in the theory of finite topologies by introducing the idea of generating topologies. Using this concept, it is possible to derive existence theorems and get numerical results in an easy manner.