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, Supporting iPython code for counting nonequivalent closure systems w.r.t. the T_1 separation axiom, Github repository
Eric Weisstein's World of Mathematics, Separation Axioms
Wikipedia, Separation Axiom
EXAMPLE
a(0) = 1 counts the empty set, while a(1) = 2 counts {{1}} and {{},{1}}.
For a(2) = 1 the closure system is as follows: {{1,2},{1},{2},{}}.
The a(3) = 4 inequivalent set-systems of closed sets are:
{{1,2,3},{1},{2},{3},{}}
{{1,2,3},{1,2},{1},{2},{3},{}}
{{1,2,3},{1,2},{1,3},{1},{2},{3},{}}
{{1,2,3},{1,2},{1,3},{2,3},{1},{2},{3},{}}.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Dmitry I. Ignatov, Jul 05 2022
STATUS
approved