A334255 - OEIS

%I #25 Sep 06 2022 14:57:59

%S 1,1,1,8,545,702525,66960965307

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

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

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

%H Dmitry I. Ignatov, <a href="https://github.com/dimachine/ClosureSeparation/">Supporting iPython code for counting closure systems w.r.t. the T_1 separation axiom</a>, Github repository

%H Dmitry I. Ignatov, <a href="/A334255/a334255.ipynb.txt">Supporting iPython notebook</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SeparationAxioms.html">Separation Axioms</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Separation_axiom">Separation Axiom</a>

%e The a(3) = 8 set-systems of closed sets:

%e   {{1,2,3},{1},{2},{3},{}}

%e   {{1,2,3},{1,2},{1},{2},{3},{}}

%e   {{1,2,3},{1,3},{1},{2},{3},{}}

%e   {{1,2,3},{2,3},{1},{2},{3},{}}

%e   {{1,2,3},{1,2},{1,3},{1},{2},{3},{}}

%e   {{1,2,3},{1,2},{2,3},{1},{2},{3},{}}

%e   {{1,2,3},{1,3},{2,3},{1},{2},{3},{}}

%e   {{1,2,3},{1,2},{1,3},{2,3},{1},{2},{3},{}}

%t Table[Length[

%t   Select[Subsets[Subsets[Range[n]]],

%t    And[MemberQ[#, {}], MemberQ[#, Range[n]],

%t      SubsetQ[#, Intersection @@@ Tuples[#, 2]],

%t      SubsetQ[#, Map[{#} &, Range[n]]]] &]], {n, 0, 4}] (* _Tian Vlasic_, Jul 29 2022 *)

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

%Y For all strict T_0 closure operators, see A334253.

%Y For T_1 closure operators, see A334254.

%Y Cf. A326960, A326961, A326979.

%K nonn,more

%O 0,4

%A _Joshua Moerman_, Apr 24 2020

%E a(6) from _Dmitry I. Ignatov_, Jul 03 2022