The number of generalized topologies on a finite set

The number T(n) of topologies on a finite set X of cardinal n is a famous open problem. There is no known simple formula to compute T(n) for arbitrary n. The online Encyclopedia of Integer Sequences presently lists T(n) for n ≤ 18. Let X be a finite set having n elements. A subset μ of the power set expX is a generalized topology (briey GT) in X iff Gi ∈ μ (i ∈ I) implies [I ∈IGi ∈ μ (in particular, I can be empty so that the defnition implies ∅ 2 μ). Let gt(n, k) be the set of all labeled generalized topologies on X having k open sets and GT(n, k) = |gt(n, k)|. We comput GT(n, k) for k ≤ 6 and 2^n − 7 ≤ k ≤ 2^n. A GT-chain on X, is a generalized topology whose open sets are totally ordered by inclusion. The totall number of GT-chains on X are computed.