چكيده فارسي :
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.
