Kuratowski's closure-complement problem

TheInfoList

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 ...
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 ...

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: :$\left(0,1\right)\cup\left(1,2\right)\cup\\cup\bigl\left(\left[4,5\right]\cap\Q\bigr\right),$ where $\left(1,2\right)$ denotes an Interval (mathematics)#Terminology, open interval and $\left[4,5\right]$ 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.