Kuratowski's closure-complement problem

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:

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

where $(1,2)$ denotes an open interval and $[4,5]$ denotes a closed interval. Let $X$ denote this set. Then the following 14 sets are accessible:

1. $X$, the set shown above.
2. $cX=(-\backslash infty,0]\backslash cup\backslash \{1\backslash \}\backslash cup[2,3)\backslash cup(3,4)\backslash cup\backslash bigl((4,5)\backslash setminus\backslash Q\backslash bigr)\backslash cup(5,\backslash infty)$
3. $kcX=(-\backslash infty,0]\backslash cup\backslash \{1\backslash \}\backslash cup[2,\backslash infty)$
4. $ckcX=(0,1)\backslash cup(1,2)$
5. $kckcX=[0,2]$
6. $ckckcX=(-\backslash infty,0)\backslash cup(2,\backslash infty)$
7. $kckckcX=(-\backslash infty,0]\backslash cup[2,\backslash infty)$
8. $ckckckcX=(0,2)$
9. $kX=[0,2]\backslash cup\backslash \{3\backslash \}\backslash cup[4,5]$
10. $ckX=(-\backslash infty,0)\backslash cup(2,3)\backslash cup(3,4)\backslash cup(5,\backslash infty)$
11. $kckX=(-\backslash infty,0]\backslash cup[2,4]\backslash cup[5,\backslash infty)$
12. $ckckX=(0,2)\backslash cup(4,5)$
13. $kckckX=[0,2]\backslash cup[4,5]$
14. $ckckckX=(-\backslash infty,0)\backslash cup(2,4)\backslash cup(5,\backslash infty)$

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.{{Cite journal

| last = Hammer

| first = P. C.

| title = Kuratowski's Closure Theorem

| journal = Nieuw Archief voor Wiskunde

| volume = 8

| publisher = Royal Dutch Mathematical Society

| pages = 74–80

| year = 1960

| issn = 0028-9825}}

The closure-complement operations yield a monoid that can be used to classify topological spaces.{{Cite journal|title=The radical-annihilator monoid of a ring|first=Ryan|last=Schwiebert|journal=Communications in Algebra |year=2017 |volume=45 |issue=4 |pages=1601–1617 |doi=10.1080/00927872.2016.1222401|arxiv=1803.00516|s2cid=73715295 }}

{{Reflist}}

• [http://nzjm.math.auckland.ac.nz/images/6/63/The_Kuratowski_Closure-Complement_Theorem.pdf The Kuratowski Closure-Complement Theorem] {{Webarchive|url=https://web.archive.org/web/20220212062843/http://nzjm.math.auckland.ac.nz/images/6/63/The_Kuratowski_Closure-Complement_Theorem.pdf |date=2022-02-12 }} by B. J. Gardner and Marcel Jackson
• [http://www.maa.org/publications/periodicals/loci/supplements/the-kuratowski-closure-complement-problem The Kuratowski Closure-Complement Problem] by Mark Bowron

Category:Topology

Category:Mathematical problems

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