In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set.
They were first formalized by Kazimierz Kuratowski,[1] and the idea was further studied by mathematicians such as Wacław Sierpiński and António Monteiro,[2] among others.
A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator.
preserving binary unions is the following condition:[4] In fact if we rewrite the equality in [K4] as an inclusion, giving the weaker axiom [K4''] (subadditivity): then it is easy to see that axioms [K4'] and [K4''] together are equivalent to [K4] (see the next-to-last paragraph of Proof 2 below).
Kuratowski (1966) includes a fifth (optional) axiom requiring that singleton sets should be stable under closure: for all
[5] If requirement [K3] is omitted, then the axioms define a Čech closure operator.
is called Kuratowski, Čech or Moore closure space depending on the axioms satisfied by
A more symmetric alternative to [M] was also proven by M. O. Botelho and M. H. Teixeira to imply axioms [K2]–[K4]:[2] A dual notion to Kuratowski closure operators is that of Kuratowski interior operator, which is a map
; 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 [K4'], and because of intensivity [I2], it is possible to weaken the equality in [I3] to a simple inclusion.
The duality between Kuratowski closures and interiors is provided by the natural complement operator on
This map is an orthocomplementation on the power set lattice, meaning it satisfies De Morgan's laws: if
by employing these relations in conjunction with the properties of the orthocomplementation
Notice that axioms [K1]–[K4] may be adapted to define an abstract unary operation
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 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
A closure operator naturally induces a topology as follows.
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
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
of all closed sets satisfies the following: [T2] It is complete under arbitrary intersections, i.e. if
The preservation of the empty set [K1] readily implies
satisfying axioms [T1]–[T3], it is possible to construct a Kuratowski closure operator in the following way: if
by axiom [T1], so the intersection collapses to the null set and [K1] follows.
of all Čech closure operators, which strictly contains
is also surjective, which signifies that all Čech closure operators on
In any induced topology (relative to the subset A) the closed sets induce a new closure operator that is just the original closure operator restricted to A:
This can be used to define a proximity relation on the points and subsets of a set.
is connected iff it cannot be written as the union of two separated subsets.