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In my topology class we are asked to list all topologies on a $3$ element set. I have found $29$ and this should be the correct result. Now I wonder whether there is some formula that determines this number exactely or at least whether there are better bounds than $2$ and $2^{2^n}$ ;)

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The sequence is OEIS A000798. Your $29$ is correct. It begins

$1, 1, 4, 29, 355, 6942, 209527, 9535241, 642779354, 63260289423, 8977053873043, 1816846038736192, 519355571065774021, 207881393656668953041, 115617051977054267807460, 88736269118586244492485121, 93411113411710039565210494095, 134137950093337880672321868725846, 261492535743634374805066126901117203$

and a formula in terms of the number of partial orders and Stirling numbers is given.

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    $\begingroup$ Exam exercise one: Write down all topologies on $\{1,2,3,4\}$ ;) $\endgroup$ Apr 24, 2015 at 15:31

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