In
point-set topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
, 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 something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
to a given starting subset of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
. The answer is 14. This result was first published by
Kazimierz Kuratowski
Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics.
Biography and studies
Kazimierz Kuratowski was born in Warsaw, (t ...
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
denote an arbitrary subset of a topological space, write
for the closure of
, and
for the complement of
. The following three identities imply that no more than 14 distinct sets are obtainable:
#
. (The closure operation is
idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
.)
#
. (The complement operation is an
involution
Involution may refer to:
* Involute, a construction in the differential geometry of curves
* '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
.)
#
. (Or equivalently
, using identity (2)).
The first two are trivial. The third follows from the identity
where
is the
interior of
which is equal to the complement of the closure of the complement of
,
. (The operation
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:
:
where
denotes an
open interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
and