Abstract
Let S denote a finite set of cardinal n. The discrete topology on S contains 2 n 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 2 n . Other bounds are derived on the cardinality of connected, non- T 0, connected and non- T 0, and non-connected topologies. Proofs involve results in the theory of transitive digraphs.
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