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 $n\leq 18$‎. ‎Let $X$ be a finite set having $n$ elements‎. ‎A subset $\mu$ of the power‎ ‎set $exp X$ is a {\it generalized topology} (briefly GT) in X iff‎ ‎$G _{i} \in \mu$ $( i \in I )$ implies $\cup_{ i\in I} G_{i} \in \mu$ (in particular‎, ‎$I$ can be empty‎ ‎so that the defnition implies $\emptyset \in \mu$)‎. ‎Let $ g t (n‎, ‎k )$ be the set‎ ‎of all labeled generalized topologies on X having k open sets and‎ ‎$GT (n‎, ‎k ) = |g t (n‎, ‎k )|$‎. ‎We comput $GT(n,k)$ for $k\leq 6$ and $2^{n}-7\leq k\leq 2^{n}$‎. ‎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.