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.
well, start with the basics, what is a topology? it is a pair of a set and a set of "open" subsets of , which we denote here, , such that we have the following axioms holding:
(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?