As far as I know there isn't an exact formula for the number of topologies on a finite set with n elements, for large n... I will apreciate any information on this topic. Thank you.
(1) arbitrary unions of open sets are open
(2) the intersection of any two open sets is open
(3) and are open
so really, counting the elements in a topology amounts to counting subsets of a set, , and the numbers of subsets we can form from the subset of the set . of course, the subset has to contain at least and .
now do you think you can answer your problem?