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(Bow). Thank you.

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- Oct 12th 2008, 11:39 AMVevethe 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(Bow). Thank you.

- Oct 12th 2008, 12:34 PMJhevon
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 $\displaystyle \emptyset$.

now do you think you can answer your problem? - Oct 12th 2008, 01:35 PMThePerfectHacker
I think the member understands the problem he is rather asking how to find the formula which seems to be an unsolved combinatorical problem. (Surprised)

Just search the internet and something appears.