Kuratowski's closure-complement problem

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] The problem gained wide exposure three decades later as an exercise in John L. Kelley's classic textbook General Topology.[2]

Proof

Letting S denote an arbitrary subset of a topological space, write kS for the closure of S, and cS for the complement of S. The following three identities imply that no more than 14 distinct sets are obtainable:

(1) kkS = kS. (The closure operation is idempotent.)

(2) ccS = S. (The complement operation is an involution.)

(3) kckckckcS = kckcS.(Or equivalently kckckckS=kckckckccS=kckS. Using identity (2).)

The first two are trivial. The third follows from the identity kikiS = kiS where iS is the interior of S which is equal to the complement of the closure of the complement of S, iS = ckcS. (The operation ki = kckc 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:

${\displaystyle (0,1)\cup (1,2)\cup \{3\}\cup {\bigl (}[4,5]\cap \mathbb {Q} {\bigr )},}$

where ${\displaystyle (1,2)}$ denotes an open interval and ${\displaystyle [4,5]}$ denotes a closed interval.

Further results

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]

References

1. ^ Kuratowski, Kazimierz (1922). "Sur l'operation A de l'Analysis Situs" (PDF). Fundamenta Mathematicae. Warsaw: Polish Academy of Sciences. 3: 182–199. ISSN 0016-2736.
2. ^ Kelley, John (1955). General Topology. Van Nostrand. p. 57. ISBN 0-387-90125-6.
3. ^ Hammer, P. C. (1960). "Kuratowski's Closure Theorem". Nieuw Archief voor Wiskunde. Royal Dutch Mathematical Society. 8: 74–80. ISSN 0028-9825.