Abstract
Every topology τ on the finite set X={x1,...,xN} has a minimal base Μ(τ); separation axioms and connectivity are simply characterized; further we obtain a recursive construction of finite topologies.
In the second section we give some formulae connecting A(N) (the number of topologies on X), Z(N) (the number of connected topologies on X) and associated numbers. The theory of representation matrices discussed in the third section leads to an easy description of many topological notions as closure operator, induced topology, connectivity; it also yields some more combinatorial formulae.
In the last part we improve the remainder term R(N)=Id A(N)−1/4(N)2=0(N3/2ld N) given by Kleitman and Rothschild to 0(N·ld2N). Then, among many other asymptotical results, we show that A(N) and Z(N) are asymptotically equal.
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Erné, M. Struktur- und anzahlformeln für topologien auf endlichen mengen. Manuscripta Math 11, 221–259 (1974). https://doi.org/10.1007/BF01173716
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DOI: https://doi.org/10.1007/BF01173716