
EXAMPLE

The a(3) = 8 setsystems of closed sets:
{{1,2,3},{1},{2},{3},{}}
{{1,2,3},{1,2},{1},{2},{3},{}}
{{1,2,3},{1,3},{1},{2},{3},{}}
{{1,2,3},{2,3},{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

Table[Length[
Select[Subsets[Subsets[Range[n]]],
And[MemberQ[#, {}], MemberQ[#, Range[n]],
SubsetQ[#, Intersection @@@ Tuples[#, 2]],
SubsetQ[#, Map[{#} &, Range[n]]]] &]], {n, 0, 4}] (* Tian Vlasic, Jul 29 2022 *)
