In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and related branches of
mathematics, the Kuratowski closure axioms are a set of
axioms that can be used to define a
topological structure
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 point ...
on a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
. They are equivalent to the more commonly used
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
definition. They were first formalized 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 ...
, and the idea was further studied by mathematicians such as
Wacław Sierpiński
Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and t ...
and
António Monteiro,
among others.
A similar set of axioms can be used to define a topological structure using only the dual notion of
interior operator.
Definition
Kuratowski closure operators and weakenings
Let
be an arbitrary set and
its
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
. A Kuratowski closure operator is a
unary operation
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation o ...
with the following properties:
A consequence of
preserving binary unions is the following condition:
In fact if we rewrite the equality in
4'' as an inclusion, giving the weaker axiom
''">4'''' (''subadditivity''):
then it is easy to see that axioms
4''' and
''">4'''' together are equivalent to
4'' (see the next-to-last paragraph of Proof 2 below).
includes a fifth (optional) axiom requiring that singleton sets should be stable under closure: for all
,
. He refers to topological spaces which satisfy all five axioms as ''T
1-spaces'' in contrast to the more general spaces which only satisfy the four listed axioms. Indeed, these spaces correspond exactly to the
topological T1-spaces via the usual correspondence (see below).
[.]
If requirement
3'' is omitted, then the axioms define a Čech closure operator. If
1'' is omitted instead, then an operator satisfying
2'',
3'' and
4''' is said to be a Moore closure operator. A pair
is called Kuratowski, Čech or Moore closure space depending on the axioms satisfied by
.
Alternative axiomatizations
The four Kuratowski closure axioms can be replaced by a single condition, given by Pervin:
Axioms
1''–
4'' can be derived as a consequence of this requirement:
# Choose
. Then
, or
. This immediately implies
1''.
# Choose an arbitrary
and
. Then, applying axiom
1'',
, implying
2''.
# Choose
and an arbitrary
. Then, applying axiom
1'',
, which is
3''.
# Choose arbitrary
. Applying axioms
1''–
3'', one derives
4''.
Alternatively, had proposed a weaker axiom that only entails
2''–
4'':
Requirement
1'' is independent of
'' : indeed, if
, the operator
defined by the constant assignment
satisfies
'' but does not preserve the empty set, since
. Notice that, by definition, any operator satisfying
'' is a Moore closure operator.
A more symmetric alternative to
'' was also proven by M. O. Botelho and M. H. Teixeira to imply axioms
2''–
4'':
[.]
Analogous structures
Interior, exterior and boundary operators
A dual notion to Kuratowski closure operators is that of Kuratowski interior operator, which is a map
satisfying the following similar requirements:
[.]
For these operators, one can reach conclusions that are completely analogous to what was inferred for Kuratowski closures. For example, all Kuratowski interior operators are ''isotonic'', i.e. they satisfy
4''', and because of intensivity
2'', it is possible to weaken the equality in
3'' to a simple inclusion.
The duality between Kuratowski closures and interiors is provided by the natural complement operator on
, the map
sending
. This map is an
orthocomplementation on the power set lattice, meaning it satisfies
De Morgan's laws
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British math ...
: if
is an arbitrary set of indices and
,
By employing these laws, together with the defining properties of
, one can show that any Kuratowski interior induces a Kuratowski closure (and vice versa), via the defining relation
(and
). Every result obtained concerning
may be converted into a result concerning
by employing these relations in conjunction with the properties of the orthocomplementation
.
further provides analogous axioms for Kuratowski exterior operators
and Kuratowski boundary operators, which also induce Kuratowski closures via the relations
and
.
Abstract operators
Notice that axioms
1''–
4'' may be adapted to define an ''abstract'' unary operation
on a general bounded lattice
, by formally substituting set-theoretic inclusion with the partial order associated to the lattice, set-theoretic union with the join operation, and set-theoretic intersections with the meet operation; similarly for axioms
1''–
4''. If the lattice is orthocomplemented, these two abstract operations induce one another in the usual way. Abstract closure or interior operators can be used to define a
generalized topology In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and or ...
on the lattice.
Since neither unions nor the empty set appear in the requirement for a Moore closure operator, the definition may be adapted to define an abstract unary operator
on an arbitrary
poset
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
.
Connection to other axiomatizations of topology
Induction of topology from closure
A closure operator naturally induces a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
as follows. Let
be an arbitrary set. We shall say that a subset
is closed with respect to a Kuratowski closure operator
if and only if it is a ''fixed point'' of said operator, or in other words it is ''stable under''
, i.e.
. The claim is that the family of all subsets of the total space that are complements of closed sets satisfies the three usual requirements for a topology, or equivalently, the family
. Then