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q question about formula topo base and open set?

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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

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

{
#1 {
{ 1, 3 },
{ 3 },
{ 1, 2 }
},
#2{},
$1{
{ 1 },
{ 2 },
{ 2, 3 },
{ 1, 2 }
},
{#3
{ 1 },
{ 3 },
{ 2, 3 }
},
{
$2 { 2 },
{ 3 },
{ 2, 3 },
{ 1, 2 }
},
{#4
{ 1 },
{ 2 },
{ 3 },
{ 1, 2 }
},
{$3
{ 2 },
{ 2, 3 }
},
{
#5 { 2, 3 },
{ 1, 2 }
},
{
#6 { 1, 3 },
{ 2, 3 }
},
{
#7 { 1 },
{ 1, 3 },
{ 2 },
{ 2, 3 }
},
{
#8{ 1 },
{ 3 }
},
{
#9{ 1 },
{ 2 },
{ 2, 3 }
},
{
$4 { 2, 3 }
},
{#10
{ 1 },
{ 2 },
{ 3 },
{ 2, 3 }
},
{#11
{ 1 },
{ 1, 3 },
{ 2 },
{ 3 },
{ 1, 2 }
},
{$5
{ 1 },
{ 1, 3 },
{ 3 },
{ 1, 2 }
},
{$6
{ 1 },
{ 1, 3 },
{ 2 },
{ 1, 2 }
},
{#12
{ 1 },
{ 1, 3 },
{ 2, 3 },
{ 1, 2 }
},
{#13
{ 3 },
{ 2, 3 },
{ 1, 2 }
},
{$7
{ 1, 2 }
},
{#13
{ 1 },
{ 1, 3 },
{ 2 },
{ 2, 3 },
{ 1, 2 }
},
{$8----------------离散
{ 1 },
{ 1, 3 },
{ 2 },
{ 3 },
{ 2, 3 },
{ 1, 2 }
},
{$9
{ 2 },
{ 3 },
{ 2, 3 }
},
{$10
{ 1, 3 },
{ 2 },
{ 3 },
{ 2, 3 },
{ 1, 2 }
},
{$11
{ 2 },
{ 2, 3 },
{ 1, 2 }
},
{$12
{ 1 },
{ 1, 3 },
{ 3 },
{ 2, 3 }
},
{#15
{ 1 },
{ 1, 3 },
{ 2, 3 }
},
{$13
{ 1, 3 },
{ 3 }
},
{#16
{ 2 },
{ 3 },
{ 1, 2 }
},
{#17
{ 1, 3 },
{ 2, 3 },
{ 1, 2 }
},
{#18
{ 1 },
{ 2 },
{ 3 }
},
{$14
{ 1 },
{ 2, 3 }
},
{#19
{ 1 },
{ 1, 3 },
{ 2 }
},
{#20
{ 1 },
{ 2 }
},
{#21
{ 1, 3 },
{ 1, 2 }
},
{$15
{ 1, 3 },
{ 2 }
},
{#22
{ 1 },
{ 3 },
{ 2, 3 },
{ 1, 2 }
},
{#23
{ 1, 3 },
{ 3 },
{ 2, 3 },
{ 1, 2 }
},
{#24
{ 1 },
{ 3 },
{ 1, 2 }
},
{#25
{ 1, 3 },
{ 2 },
{ 3 },
{ 1, 2 }
},
{#26
{ 1 },
{ 2, 3 },
{ 1, 2 }
},
{$16
{ 1 },
{ 1, 2 }
},
{$17
{ 2 },
{ 1, 2 }
},
{$18
{ 3 },
{ 1, 2 }
},
{#27
{ 1, 3 },
{ 2 },
{ 2, 3 },
{ 1, 2 }
},
{$19
{ 1, 3 }
},
{#28
{ 1, 3 },
{ 2 },
{ 2, 3 }
},
{$20
{ 2 }
},
{#29
{ 1 },
{ 2 },
{ 3 },
{ 2, 3 },
{ 1, 2 }
},
{#21
{ 1 }
},
{#22
{ 1 },
{ 2 },
{ 1, 2 }
},
{$23
{ 1 },
{ 1, 3 }
},
{$24
{ 1 },
{ 1, 3 },
{ 3 }
},
{#30
{ 1 },
{ 1, 3 },
{ 2 },
{ 3 },
{ 2, 3 }
},
{$25
{ 3 },
{ 2, 3 }
},
{$26
{ 1, 3 },
{ 3 },
{ 2, 3 }
},
{#31
{ 1, 3 },
{ 2 },
{ 1, 2 }
},
{$27
{ 1, 3 },
{ 2 },
{ 3 },
{ 2, 3 }
},
{#32
{ 1 },
{ 1, 3 },
{ 2 },
{ 3 }
},
{#33
{ 2 },
{ 3 }
},
{#34
{ 1, 3 },
{ 2 },
{ 3 }
},
{$28
{ 3 }
},
{$29
{ 1 },
{ 1, 3 },
{ 1, 2 }
},
{#35
{ 1 },
{ 1, 3 },
{ 3 },
{ 2, 3 },
{ 1, 2 }
}
}
64 =29$+35#


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

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.

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