# Thread: the number of topologies on a finite set

1. ## 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.

2. 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 $(X, \mathcal{O})$ of a set $X$ and a set of "open" subsets of $X$, which we denote here, $\mathcal{O}$, 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) $\emptyset$ and $X$ are open

so really, counting the elements in a topology amounts to counting subsets of a set, $X$, and the numbers of subsets we can form from the subset of the set $X$. of course, the subset has to contain at least $X$ and $\emptyset$.

3. Originally Posted by Jhevon
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.