# q question about formula topo base and open set?

is there a formula about calculating all open set in collection(set)?

I try {a,b,c},29 open set by topo base ,take more 2 hours........

if {a,b,c,d},maybe a week....

below small order set,but what is formula?

http://oeis.org/A000798

Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.
(formerly M3631 N1476)

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

{
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{ 1, 3 },
{ 2 },
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{ 1, 2, 3 },
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}
8
=====================
{
{ 1 },
{ 1, 3 },
{ 2 },
{ 3 },
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{ 1, 2 }
}
6
=======================

{
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{#4
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{#10
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{#11
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{#12
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{#13
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{$9 { 2 }, { 3 }, { 2, 3 } }, {$10
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{$11 { 2 }, { 2, 3 }, { 1, 2 } }, {$12
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{#15
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{$13 { 1, 3 }, { 3 } }, {#16 { 2 }, { 3 }, { 1, 2 } }, {#17 { 1, 3 }, { 2, 3 }, { 1, 2 } }, {#18 { 1 }, { 2 }, { 3 } }, {$14
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{#19
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{#20
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{#27
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{$19 { 1, 3 } }, {#28 { 1, 3 }, { 2 }, { 2, 3 } }, {$20
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{#29
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{#31
{ 1, 3 },
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{$27 { 1, 3 }, { 2 }, { 3 }, { 2, 3 } }, {#32 { 1 }, { 1, 3 }, { 2 }, { 3 } }, {#33 { 2 }, { 3 } }, {#34 { 1, 3 }, { 2 }, { 3 } }, {$28
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},
{$29 { 1 }, { 1, 3 }, { 1, 2 } }, {#35 { 1 }, { 1, 3 }, { 3 }, { 2, 3 }, { 1, 2 } } } 64 =29$+35#

### Re: q question about formula topo base and open set?

What you want to say is that there is no formula which can be expressed in the terms of some canonical functions like polynomials or exponentials/logarithms or any other elementary functions. Obviously the number of topologies T(n) on a set with n elements can be bounded by

T(n) < 2^(2^n)

and there are many interesting results on this topic. See for example:

http://en.wikipedia.org/wiki/Finite_topological_space

joking

### Re: q question about formula topo base and open set?

There is no formula in terms of n=number of points, but of course, this should be proven, which I don't believe is the case.