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: :

cap\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,4 cup,\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