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Kuratowski's closure-complement problem

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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

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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)∪(1,2)∪{3}∪([4,5]∩ℚ),

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=(−∞,0]∪{1}∪[2,3)∪(3,4)∪((4,5)∖ℚ)∪(5,∞)
  3. kcX=(−∞,0]∪{1}∪[2,∞)
  4. ckcX=(0,1)∪(1,2)
  5. kckcX=[0,2]
  6. ckckcX=(−∞,0)∪(2,∞)
  7. kckckcX=(−∞,0]∪[2,∞)
  8. ckckckcX=(0,2)
  9. kX=[0,2]∪{3}∪[4,5]
  10. ckX=(−∞,0)∪(2,3)∪(3,4)∪(5,∞)
  11. kckX=(−∞,0]∪[2,4]∪[5,∞)
  12. ckckX=(0,2)∪(4,5)
  13. kckckX=[0,2]∪[4,5]
  14. ckckckX=(−∞,0)∪(2,4)∪(5,∞)

Further results

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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

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References

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  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. {{cite book}}: ISBN / Date incompatibility (help)
  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. arXiv:2210.10928. 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.
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