Thread by @SamuelGWalters

For any subset A of a topological space X, there are no more than 14 sets that can be formed from A using the closure and complement operations. The complement of B is B' = X-B. Closure C(B) = smallest closed set containing B. (Neat problem from Kelley's General #Topology.) #math

There's a lemma that helps one show this. Letting C(A) denote the closure of A, and N(A) the complement of A, we can prove:

Lemma: for all sets A, one has the equation
CNC N CNC(A) = CNC(A).

The result follows from the semigroup with identity I generated by 2 elements N and C such that N² = I, C² = C, and CNCNCNC = CNC. Its 14 elements are:
I N C NC CN NCN CNC CNCN NCNC
NCNCN CNCNC CNCNCN NCNCNC NCNCNCN.

Should have nice matrix representation. #math

Fairly self-contained proof of a lemma in #topology useful in proving an earlier result (that there are at most 14 sets that can be formed from a given set using closures and complements). #math #geometry