Kuratowski's closure-complement problem
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In
point-set topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
, Kuratowski's closure-complement problem asks for the largest number of distinct sets obtainable by repeatedly applying the set operations of closure and
complement A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
to a given starting subset of a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a ...
. The answer is 14. This result was first published by
Kazimierz Kuratowski Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish Polish may refer to: * Anything from or related to Poland Poland ( pl, Polska ), officially the Republic of Poland ( pl, Rzeczpospolita Polska, links=no ), is a cou ...

Kazimierz Kuratowski
in 1922. The problem gained wide exposure three decades later as an exercise in John L. Kelley's classic textbook ''General Topology''.


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 idempotency, idempotent.) # ''ccS'' = ''S''. (The complement operation is an involution (mathematics), 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 (topology), 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\\cup\bigl([4,5]\cap\Q\bigr), where (1,2) denotes an Interval (mathematics)#Terminology, open interval and [4,5] denotes a closed interval.


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. The closure-complement operations yield a monoid that can be used to classify topological spaces.


References


External links


The Kuratowski Closure-Complement Theorem
by B. J. Gardner and Marcel Jackson
The Kuratowski Closure-Complement Problem
by Mark Bowron Topology Mathematical problems {{topology-stub