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 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 (mathematics), distance. More specifically, a to ...
. The answer is 14. This result was first published by Kazimierz Kuratowski in 1922. 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''.


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 In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...
under the usual topology contains 14-sets. Here is one example: :(0,1)\cup(1,2)\cup\\cup\bigl( ,5cap\Q\bigr), where (1,2) denotes an
open interval In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
and ,5/math> denotes a closed interval. Let X denote this set. Then the following 14 sets are accessible: # X, the set shown above. # cX=(-\infty,0]\cup\\cup ,3)\cup(3,4)\cup\bigl((4,5)\setminus\Q\bigr)\cup(5,\infty) # kcX=(-\infty,0cup\\cup ,\infty) # ckcX=(0,1)\cup(1,2) # kckcX=[0,2/math> # ckckcX=(-\infty,0)\cup(2,\infty) # kckckcX=(-\infty,0]\cup[2,\infty) # ckckckcX=(0,2) # kX=[0,2]\cup\\cup ,5/math> # ckX=(-\infty,0)\cup(2,3)\cup(3,4)\cup(5,\infty) # kckX=(-\infty,0]\cup ,4cup ,\infty) # ckckX=(0,2)\cup(4,5) # kckckX=[0,2cup ,5/math> # 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. 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