Kuratowski's closure-complement problem

In 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. In 1955 John Kelley popularized the problem by including it as an exercise in his widely-adopted textbook on topology. A surprisingly large number of related problems and results have appeared since 1960.

Letting S denote an arbitrary subset of a topological space, and writing kS for the closure of S, iS for the interior of S, and cS for the complement of S, Kuratowski's 14-set result follows easily from these three facts:

(1) kkS = kS

(2) ccS = S

(3) kckckckS = kckS.

The first two are trivial. The third follows from the identities kikiS = kiS and iS = ckcS.

A subset realizing the maximum of 14 is called a 14-set. The set of real numbers under the usual topology contains 14-sets. Here is one example:

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

where $(0,1)$ and $(1,2)$ denote open intervals and $[4,5]$ denotes a closed interval.

External links

• The Kuratowski Closure-Complement Theorem by B.J. Gardner and Marcel Jackson