OFFSET
0,2
COMMENTS
The T_1 axiom states that all singleton sets {x} are closed.
For n>1, this property implies strictness (meaning that the empty set is closed).
LINKS
Dmitry I. Ignatov, On the Cryptomorphism between Davis' Subset Lattices, Atomic Lattices, and Closure Systems under T1 Separation Axiom, arXiv:2209.12256 [cs.DM], 2022.
Dmitry I. Ignatov, Supporting iPython code for counting closure systems w.r.t. the T_1 separation axiom, Github repository
Dmitry I. Ignatov, PDF of the supporting iPython notebook
S. Mapes, Finite atomic lattices and resolutions of monomial ideals, J. Algebra, 379 (2013), 259-276.
Eric Weisstein's World of Mathematics, Separation Axioms
Wikipedia, Separation Axiom
EXAMPLE
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},{}}
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
Joshua Moerman, Apr 20 2020
EXTENSIONS
a(6) from Dmitry I. Ignatov, Jul 03 2022
STATUS
approved