the number of topologies on a finite set

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October 12th 2008, 11:39 AM
Veve

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(Bow). Thank you.
October 12th 2008, 12:34 PM
Jhevon

Quote:

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(Bow). 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 $\emptyset$ .

now do you think you can answer your problem?
October 12th 2008, 01:35 PM
ThePerfectHacker

Quote:

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. (Surprised)

Quote:

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(Bow). Thank you.

Just search the internet and something appears.