Kuratowski's closure-complement problem

Kuratowski's closure-complement problem asks how many distinct sets repeated closure and complementation can produce from one subset of a topological space. The maximum is 14.

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Closure operator: A function assigning each subset its smallest closed superset, satisfying extensivity, monotonicity, and idempotence. Its idempotence is one of the relations that limits repeated operations.
Kazimierz Kuratowski: A Polish mathematician whose work helped establish modern topology and set theory. Kuratowski formulated the closure-complement problem and proved its 14-set bound.
Topological space: A set equipped with a topology, a collection of subsets designated as open. Closure is defined relative to the topology on the underlying space.
Real line: The set of real numbers with its usual topology. A suitable subset of the real line yields the full 14-set orbit.
Kuratowski's 14-set theorem: The result that closure and complementation generate no more than 14 sets from one subset. This names the established theorem, while the problem emphasizes finding or understanding the maximum.
Interior operator: A function assigning each subset its largest open subset, satisfying contractivity, monotonicity, and idempotence. Complementation turns closure into interior, so both operations generate the same family.
Topology: The study of properties of spaces preserved under continuous deformation. The problem abstracts a basic topological operation into an algebraic question.
Open set: A member of a topology, characterized by the neighborhood condition at each of its points. Complementation converts closed sets into open sets during iteration.
Rational numbers: The numbers expressible as ratios of integers with nonzero denominator. Their density in the real line makes them a useful starting subset for closure examples.
Boolean algebra: An algebraic structure with conjunction, disjunction, and complementation satisfying Boolean laws. Complementation obeys Boolean laws, but closure adds topological constraints absent from a Boolean algebra.