A334255 Number of strict closure operators on a set of n elements which satisfy the T_1 separation axiom.

1, 1, 1, 8, 545, 702525, 66960965307

Offset: 0

Author

Joshua Moerman, Apr 24 2020

Keywords

• nonn
• more

Comments

The T_1 axiom states that all singleton sets {x} are closed.

A closure operator is strict if the empty set is closed.

Examples

			The a(3) = 8 set-systems 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},{}}
		

Links

• Dmitry I. Ignatov, Supporting iPython code for counting closure systems w.r.t. the T_1 separation axiom, Github repository
• Dmitry I. Ignatov, Supporting iPython notebook
• Eric Weisstein's World of Mathematics, Separation Axioms
• Wikipedia, Separation Axiom

Crossrefs

The number of all strict closure operators is given in A102894.

For all strict T_0 closure operators, see A334253.

For T_1 closure operators, see A334254.

Cf. A326960, A326961, A326979.

Programs

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 *)

Extensions

a(6) from Dmitry I. Ignatov, Jul 03 2022