A396537 - OEIS

%I #13 Jun 02 2026 18:23:10

%S 1,1,1,2,18,1767

%N Number of inequivalent strict closure operators on a set of n elements such that all pairs of distinct points can be separated by clopen sets.

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

%C Two distinct points x,y in X are separated by a set H if x is an element of H and y is not an element of H.

%C Also the number of non-isomorphic S_2 convexities on a set of n elements in the sense of Chepoi.

%D G. M. Bergman. Lattices, Closure Operators, and Galois Connections. Springer, Cham. 2015. 173-212 in "An Invitation to General Algebra and Universal Constructions", Springer, (2015).

%H Victor Chepoi, <a href="https://www.researchgate.net/publication/2407147_Separation_Of_Two_Convex_Sets_In_Convexity_Structures">Separation of Two Convex Sets in Convexity Structures</a>

%H Tian Vlasic, <a href="/A396537/a396537.py.txt">Python program</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Closure_operator">Closure operator</a>

%e a(3)=2 as there are 2 distinct S_2 convexities on a set of three elements. If we set X={a,b,c}, then the two convexities could be e.g.,

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

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

%Y Cf. A358144, A334255, A358152, A356544.

%K nonn,more

%O 0,4

%A _Tian Vlasic_, May 28 2026