
EXAMPLE

The a(3) = 8 setsystems of closed sets:
{{}, {1, 2, 3}}
{{}, {1}, {2, 3}, {1, 2, 3}}
{{}, {2}, {1, 3},{1, 2, 3}}
{{}, {3}, {1, 2}, {1, 2, 3}}
{{}, {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[
Flatten[(x > ({x, #} & /@ Select[F, FreeQ[#, x] &])) /@ Range[n],
1], MemberQ[F,
_?(H > With[{H1 = Complement[Range[n], H]},
MemberQ[F, H1] && MemberQ[H, #[[1]]
] && SubsetQ[H1, #[[2]]
]])]&];
Table[Length@Select[Select[
Subsets[Subsets[Range[n]]],
And[
MemberQ[#, {}],
MemberQ[#, Range[n]],
SubsetQ[#, Intersection @@@ Tuples[#, 2]]] &
], SeparatedPairQ[#, n] &], {n, 0, 4}]
