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. The answer is 14. 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. [2]
Letting denote an arbitrary subset of a topological space, write for the closure of , and for the complement of . The following three identities imply that no more than 14 distinct sets are obtainable:
The first two are trivial. The third follows from the identity where is the interior of which is equal to the complement of the closure of the complement of , . (The operation is idempotent.)
A subset realizing the maximum of 14 is called a 14-set. The space of real numbers under the usual topology contains 14-sets. Here is one example:
where denotes an open interval and denotes a closed interval. Let denote this set. 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. [4]
{{cite book}}
: ISBN / Date incompatibility (help)