imported>Mathematrucker: /* See also */ If Namioka's theorem belongs in a See Also list on this page then so does every general topology page in Wikipedia.

2025-03-06T16:15:55Z

See also: If Namioka's theorem belongs in a See Also list on this page then so does every general topology page in Wikipedia.

New page

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 (topology)|closure]] and [[complement (set theory)|complement]] to a given starting subset of a [[topological space]]. The answer is 14. This result was first published by [[Kazimierz Kuratowski]] in 1922.<ref>{{Cite journal
| last = Kuratowski
| first = Kazimierz
| authorlink = Kazimierz Kuratowski
| title = Sur l'operation A de l'Analysis Situs
| url = http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3121.pdf
| journal = Fundamenta Mathematicae
| volume = 3
| pages = 182–199
| publisher = Polish Academy of Sciences
| location = Warsaw
| year = 1922
| doi = 10.4064/fm-3-1-182-199
| issn = 0016-2736}}</ref> It gained additional exposure in Kuratowski's fundamental monograph ''Topologie'' (first published in French in 1933; the first English translation appeared in 1966) before achieving fame as a textbook exercise in [[John L. Kelley]]'s 1955 classic, ''General Topology''.<ref>{{Cite book
| last = Kelley
| first = John
| isbn = 0-387-90125-6
| authorlink = John L. Kelley
| title = General Topology
| publisher = Van Nostrand
| year = 1955
| page = 57}}</ref>

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

# <math>kkS=kS</math>. (The closure operation is [[idempotency|idempotent]].)
# <math>ccS=S</math>. (The complement operation is an [[involution (mathematics)|involution]].)
# <math>kckckckcS=kckcS</math>. (Or equivalently <math>kckckckS=kckckckccS=kckS</math>, using identity (2)).

The first two are trivial. The third follows from the identity <math>kikiS=kiS</math> where <math>iS</math> is the [[interior (topology)|interior]] of <math>S</math> which is equal to the complement of the closure of the complement of <math>S</math>, <math>iS=ckcS</math>. (The operation <math>ki=kckc</math> 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:
:<math>(0,1)\cup(1,2)\cup\{3\}\cup\bigl([4,5]\cap\Q\bigr),</math>
where <math>(1,2)</math> denotes an [[Interval (mathematics)#Definitions|open interval]] and <math>[4,5]</math> denotes a closed interval. Let <math>X</math> denote this set. Then the following 14 sets are accessible:

# <math>X</math>, the set shown above.
# <math>cX=(-\infty,0]\cup\{1\}\cup[2,3)\cup(3,4)\cup\bigl((4,5)\setminus\Q\bigr)\cup(5,\infty)</math>
# <math>kcX=(-\infty,0]\cup\{1\}\cup[2,\infty)</math>
# <math>ckcX=(0,1)\cup(1,2)</math>
# <math>kckcX=[0,2]</math>
# <math>ckckcX=(-\infty,0)\cup(2,\infty)</math>
# <math>kckckcX=(-\infty,0]\cup[2,\infty)</math>
# <math>ckckckcX=(0,2)</math>
# <math>kX=[0,2]\cup\{3\}\cup[4,5]</math>
# <math>ckX=(-\infty,0)\cup(2,3)\cup(3,4)\cup(5,\infty)</math>
# <math>kckX=(-\infty,0]\cup[2,4]\cup[5,\infty)</math>
# <math>ckckX=(0,2)\cup(4,5)</math>
# <math>kckckX=[0,2]\cup[4,5]</math>
# <math>ckckckX=(-\infty,0)\cup(2,4)\cup(5,\infty)</math>

==Further results==
Despite its origin within the context of a topological space, Kuratowski's closure-complement problem is actually more [[algebra]]ic 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.<ref>{{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}}</ref>

The closure-complement operations yield a [[monoid]] that can be used to classify topological spaces.<ref>{{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 }}</ref>

==References==
{{Reflist}}

== External links ==
* [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]]

{{topology-stub}}

Kuratowski's closure-complement problem - Revision history

imported>Mathematrucker: /* See also */ If Namioka's theorem belongs in a See Also list on this page then so does every general topology page in Wikipedia.