Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6

The Number of Topologies on a Finite Set

Moussa Benoumhani
Faculty of Science
Department of Mathematics
Sana'a University
P.O. Box 14026
Sana'a
Yemen

Abstract: Let X be a finite set having n elements. How many different labeled topologies one can define on X? Let T(n,k) be the number of topologies having k open sets. We compute T(n,k) for 2 <= k <= 12, as well as other results concerning T₀ topologies on X having n+4 <= k <= n+6 open sets.

(Concerned with sequences A000798 A001930 and A008277 .)

Received August 29 2005; revised version received May 20 2006. Published in Journal of Integer Sequences May 24 2006.