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Math Help - the number of topologies on a finite set

  1. #1
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    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.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Veve View Post
    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.

    now do you think you can answer your problem?
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  3. #3
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    Quote Originally Posted by Jhevon View Post
    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.

    Quote Originally Posted by Veve View Post
    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.
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