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] 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. ^[2]

Proof

Letting ${\displaystyle S}$ denote an arbitrary subset of a topological space, write ${\displaystyle kS}$ for the closure of ${\displaystyle S}$, and ${\displaystyle cS}$ for the complement of ${\displaystyle S}$. The following three identities imply that no more than 14 distinct sets are obtainable:

1. ${\displaystyle kkS=kS}$. (The closure operation is idempotent.)
2. ${\displaystyle ccS=S}$. (The complement operation is an involution.)
3. ${\displaystyle kckckckcS=kckcS}$. (Or equivalently ${\displaystyle kckckckS=kckckckccS=kckS}$, using identity (2)).

The first two are trivial. The third follows from the identity ${\displaystyle kikiS=kiS}$ where ${\displaystyle iS}$ is the interior of ${\displaystyle S}$ which is equal to the complement of the closure of the complement of ${\displaystyle S}$, ${\displaystyle iS=ckcS}$. (The operation ${\displaystyle ki=kckc}$ 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. Let ${\displaystyle X}$ denote this set. Then the following 14 sets are accessible:

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

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]

Regarded as set-valued set functions, the 14 closure-complement operations comprise an operator monoid called the Kuratowski monoid where the monoid product is function composition. ^{[4] [5]} This monoid, which can be used to classify topological spaces based on the quotient they satisfy, has inspired similar classifications in other settings as well. ^[6]

See also

• Free monoid

References

1. Kuratowski, Kazimierz (1922). "Sur l'operation A de l'Analysis Situs" (PDF). Fundamenta Mathematicae. 3. Warsaw: Polish Academy of Sciences: 182–199. doi:10.4064/fm-3-1-182-199. ISSN   0016-2736.
2. Kelley, John (1955). General Topology. Van Nostrand. p. 57. ISBN   0-387-90125-6.
3. Hammer, P. C. (1960). "Kuratowski's Closure Theorem". Nieuw Archief voor Wiskunde. 8. Royal Dutch Mathematical Society: 74–80. ISSN   0028-9825.
4. Gardner, B. J.; Jackson, M. (2008). "The Kuratowski closure-complement theorem" (PDF). New Zealand Journal of Mathematics. 38. New Zealand Mathematical Society: 9–44. S2CID   11278456.
5. Bowron, Mark (2024). "Boundary-border extensions of the Kuratowski monoid". Topology and its Applications. 341 (108703). Elsevier: 1–35. doi:10.1016/j.topol.2023.108703. S2CID   257505589.
6. Schwiebert, Ryan (2017). "The radical-annihilator monoid of a ring". Communications in Algebra. 45 (4): 1601–1617. arXiv: 1803.00516 . doi:10.1080/00927872.2016.1222401. S2CID   73715295.

External links

• The Kuratowski Closure-Complement Problem Archived 2019-05-07 at the Wayback Machine