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. It follows easily from the following facts that hold for any subsets S of a space, writing S^{^} for the closure of S, S^o for the interior of S, and S' for its complement:

(1) S^{^ ^} = S^{^}

(2) S‘‘ = S

(3) S^{^} ' ^{^} ' ^{^} ' ^{^} = S^{^} ' ^{^}

The first two are trivial. (3) follows easily from the fact that S^{o ^ o ^} = S^{o ^} (together with the triviality that S^o = S‘ ^{^} ‘ ).

Many variations have appeared since, especially after 1960.

A subset realizing the maximum of 14 is called a 14-set. The real numbers in their standard topology have subsets that are 14-sets. One such subset is:

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

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

References

• Kelley, J. L. General Topology. Princeton: Van Nostrand, p. 57, 1955.
• Kuratowski, K. Sur l'operation A de l'analysis situs. Fund. Math. 3, 182-199, 1922.

External links

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