
EXAMPLE

The a(3) = 4 setsystems of closed sets:
{{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {1, 2, 3}}
{{}, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}}
{{}, {1}, {2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
{{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}


MATHEMATICA

SeparatedPairQ[F_, n_] := AllTrue[
Subsets[Range[n], {2}],
MemberQ[F,
_?(H > With[{H1 = Complement[Range[n], H]},
MemberQ[F, H1] && MemberQ[H, #[[1]]
] && MemberQ[H1, #[[2]]
]])] &];
Table[Length@Select[Select[
Subsets[Subsets[Range[n]]],
And[
MemberQ[#, {}],
MemberQ[#, Range[n]],
SubsetQ[#, Intersection @@@ Tuples[#, 2]]] &
], SeparatedPairQ[#, n] &] , {n, 0, 4}]
