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

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