Thread: the number of topologies on a finite set

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.

Originally Posted by Veve

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?

Originally Posted by Jhevon

now do you think you can answer your problem?

I think the member understands the problem he is rather asking how to find the formula which seems to be an unsolved combinatorical problem.

Originally Posted by Veve

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.

Just search the internet and something appears.