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]

The closure-complement operations yield a monoid that can be used to classify topological spaces.^[4]

References[]

External links[]

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

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