Kuratowski's closure-complement problem explained

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

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:kkS = kS . (The closure operation is idempotent.)ccS = S . (The complement operation is an involution.)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 )\cup( 1 , 2 )\cup\{ 3 \}\cup l ([ 4 , 5 ]\cap\Q r ), 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:X , the set shown above.cX = ( -inf ty, 0 ]\cup\{ 1 \}\cup[ 2 , 3 )\cup( 3 , 4 )\cup l (( 4 , 5 )\setminus\Q r )\cup( 5 , inf ty) kcX = ( -inf ty, 0 ]\cup\{ 1 \}\cup[ 2 , inf ty) ckcX = ( 0 , 1 )\cup( 1 , 2 ) kckcX = [ 0 , 2 ] ckckcX = ( -inf ty, 0 )\cup( 2 , inf ty) kckckcX = ( -inf ty, 0 ]\cup[ 2 , inf ty) ckckckcX = ( 0 , 2 ) kX = [ 0 , 2 ]\cup\{ 3 \}\cup[ 4 , 5 ] ckX = ( -inf ty, 0 )\cup( 2 , 3 )\cup( 3 , 4 )\cup( 5 , inf ty) kckX = ( -inf ty, 0 ]\cup[ 2 , 4 ]\cup[ 5 , inf ty) ckckX = ( 0 , 2 )\cup( 4 , 5 ) kckckX = [ 0 , 2 ]\cup[ 4 , 5 ] ckckckX = ( -inf ty, 0 )\cup( 2 , 4 )\cup( 5 , inf ty)

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]

External links

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

Notes and References

1. Kuratowski . Kazimierz . Kazimierz Kuratowski . Sur l'operation A de l'Analysis Situs . Fundamenta Mathematicae . 3 . 182–199 . Polish Academy of Sciences . Warsaw . 1922 . 10.4064/fm-3-1-182-199 . 0016-2736.
2. Book: Kelley , John . John L. Kelley

   . 0-387-90125-6 . John L. Kelley . General Topology . Van Nostrand . 1955 . 57.

3. Hammer . P. C. . Kuratowski's Closure Theorem . Nieuw Archief voor Wiskunde . 8 . Royal Dutch Mathematical Society . 74–80 . 1960 . 0028-9825.
4. The radical-annihilator monoid of a ring. Ryan. Schwiebert. Communications in Algebra . 2017 . 45 . 4 . 1601–1617 . 10.1080/00927872.2016.1222401. 1803.00516. 73715295 .