Cardinality of finite topologies

AbstractLet S denote a finite set of cardinal n. The discrete topology on S contains 2n open sets; the indiscrete topology contains 2 open sets. A partial answer is given to the question: For which intermediate integers m is there a topology on S with cardinal m? It is shown that no topology, other than the discrete, has cardinal greater than 3/4 2n. Other bounds are derived on the cardinality of connected, non-T0, connected and non-T0, and non-connected topologies. Proofs involve results in the theory of transitive digraphs