Direct and Elementary Approach to Enumerate Topologies on a Finite Set

Abstract:

Let $\mathbb{E}$ be a set with

elements, and let $\tau (n,k)$ be the set of all labelled topologies on $\mathbb{E}$ , having

open sets, and $T(n,k)=\left\vert \tau (n,k)\right\vert$ . In this paper, we use a direct approach to compute

for all $n\geq 4$ and $k\geq 6\cdot 2^{n-4}$ .

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(Concerned with sequences A000798, A001930, and A008277 .)

Received April 19 2006; revised version received February 28 2007. Published in Journal of Integer Sequences March 19 2007.