In point-set topology, Kuratowski's closure-complement problem asks for the largest number of distinct sets obtainable by repeatedly applying the set operations of closure and complement to a given starting subset of a topological space.
This result was first published by Kazimierz Kuratowski in 1922.
[1] It gained additional exposure in Kuratowski's fundamental monograph Topologie (first published in French in 1933; the first English translation appeared in 1966) before achieving fame as a textbook exercise in John L. Kelley's 1955 classic, General Topology.
denote an arbitrary subset of a topological space, write
The following three identities imply that no more than 14 distinct sets are obtainable: The first two are trivial.
A subset realizing the maximum of 14 is called a 14-set.
The space of real numbers under the usual topology contains 14-sets.
denotes an open interval and
Then the following 14 sets are accessible: Despite its origin within the context of a topological space, Kuratowski's closure-complement problem is actually more algebraic than topological.
A surprising abundance of closely related problems and results have appeared since 1960, many of which have little or nothing to do with point-set topology.
[3] The closure-complement operations yield a monoid that can be used to classify topological spaces.