23 11 Article 10.3.5 Journal of Integer Sequences, Vol. 13 (2010), 2 3 6 1 47 Finite Topologies and Partitions Moussa Benoumhani 1 and Messaoud Kolli Faculty of Science Department of Mathematics King Khaled University Abha Saudi Arabia benoumhani@yahoo.com kmessaud@kku.edu.sa Abstract Let E be a set with n elements, and let T (n, k) be the number of all labeled topologies having k open sets that can be defined on E. In this paper, we compute these numbers for k ≤ 17, and arbitrary n, as well as t N 0 (n, k), the number of all unlabeled non-T 0 topologies on E with k open sets, for 3 ≤ k ≤ 8. 1 Introduction Let E be an n-element set, and let T (n) be the total number of labeled topologies one can define on E. Using the one-to-one correspondence between finite topologies and idempotent 0–1-matrices with 1 in the diagonal (see Krishnamurty [17]), it is possible to establish explicit formulae for T (n) (see Ern´ e[7]), but they are too complex for a numerical evaluation in reasonable time, even on a modern computer. Comtet [5], Evans et al. [11], Renteln [19] and others reduced the computation of the numbers T (n) to that of the numbers γ n of labeled acyclic transitive graphs, in other words, of partially ordered sets with n points. These numbers are connected via the formula T (n)= ∑ n l=1 S n,l γ l , where the S n,l are the Stirling numbers of the second kind. That formula is evident in view of the one-to-one correspondence between quasiorders and topologies, respectively, between partial orders and T 0 -topologies, observed by Alexandrov [1] already in the 1930’s, and pursued later by many authors (see, e.g., [2, 3, 7, 8, 9, 10, 14]). Consequently, γ n is simultaneously the number T 0 (n) of all 1 Author’s current address: Department of Mathematics, Al-Imam University, Faculty of Sciences, P. O. Box 90950, Riyadh 11623, Saudi Arabia. 1