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 are equivalent to the more commonly used open set definition. They were first formalized by Kazimierz Kuratowski, and the idea was further studied by mathematicians such as Wacław Sierpiński and António Monteiro, among others.

A similar set of axioms can be used to define a topological structure using only the dual notion of an interior operator.

Definition

Kuratowski closure operators and weakenings

its power set. A Kuratowski closure operator is a unary operation with the following properties:

preserving binary unions is the following condition: In fact if we rewrite the equality in [K4] as an inclusion, giving the weaker axiom ] (subadditivity): then it is easy to see that axioms [K4'] and ] together are equivalent to [K4] (see the next-to-last paragraph of Proof 2 below).

. He refers to topological spaces which satisfy all five axioms as T1-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 [K3] is omitted, then the axioms define a Čech closure operator. If [K1] is omitted instead, then an operator satisfying [K2], [K3] and [K4'] is said to be a Moore closure operator. is called Kuratowski, Čech or Moore closure space.

Alternative axiomatizations

The four Kuratowski closure axioms can be replaced by a single condition, given by Pervin:

Axioms [K1]–[K4] can be derived as a consequence of this requirement:

1. . This immediately implies [K1].

2. . Then, applying axiom [K1], implying [K2].

3. . Then, applying axiom [K1], which is [K3].

4. . Applying axioms [K1]–[K3], one derives [K4].

Alternatively, had proposed a weaker axiom that only entails [K2]–[K4]: Requirement [K1] is independent of [M] satisfies [M] but does not preserve the empty set. Notice that, by definition, any operator satisfying [M] is a Moore closure operator.

A more symmetric alternative to [M] was also proven by M. O. Botelho and M. H. Teixeira to imply axioms [K2]–[K4]:

Analogous structures

Interior, exterior and boundary operators

A dual notion to Kuratowski closure operators is that of Kuratowski interior operator 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 [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. This map is an orthocomplementation on the power set lattice, meaning it satisfies De Morgan's laws

.

further provides analogous axioms for Kuratowski exterior operators and Kuratowski boundary operators,.

Abstract operators

Notice that axioms [K1]–[K4] may be adapted to define an abstract, 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 [I1]–[I4]. 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.

on an arbitrary poset.

Connection to other axiomatizations of topology

Induction of topology from closure

A closure operator naturally induces a topology is closed if and only if it is a fixed point of said operator, or in other words it is stable under of all closed sets satisfies the following:

Notice that, by idempotency [K3].

[T1] By extensivity [K2]. The preservation of the empty set [K1].

[T2] . By extensivity [K2]. Also, by isotonicity [K4'].

[T3] . From the preservation of binary unions [K4], and using induction.

Induction of closure from topology

satisfying axioms [T1]–[T3] is the inclusion upset

.

[K1] by axiom [T1], so the intersection collapses to the null set and [K1] follows.

[K2] must be contained in the intersection of all such sets. Hence follows extensivity [K2].

[K3], which is idempotence [K3].

[K4'], which is isotonicity [K4'].

[K4]. Axiom [T2]; furthermore, axiom [T2]. By extensivity [K2]. Point 4. ensures additivity [K4].

Exact correspondence between the two structures

is the collection of all families consisting of complements of all sets in a topology, i.e. the collection of all families satisfying [T1]–[T3] is a bijection.

satisfies this description: by extensivity [K2], and by idempotence [K3]: by isotonicity [K4'].

, and the latter is complete under arbitrary intersections by [T2].

. is no longer a bijection.

Examples

• is the smallest closed set is a Kuratowski closure operator.

• are Kuratowski closures. The first induces the indiscrete topology, while the second induces the discrete topology.

• , as can be seen from the definitions).

• is an infinite cardinal numbersatisfies all four Kuratowski axioms., this operator induces the cofinite topology, it induces the cocountable topology.

Properties

• Since any Kuratowski closure is isotonic, and so is obviously any inclusion mapping, one has the (isotonic) Galois connection.

• .

Topological concepts in terms of closure

Refinements and subspaces

. is an antitonic mapping between posets.

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.

Continuous maps, closed maps and homeomorphisms

is continuous. is a closed map iff the reverse inclusion holds, and it is a homeomorphism iff it is both continuous and closed, i.e. iff equality holds.

Separation axioms

be a Kuratowski closure space. Then

• is a T0-space;

• is a T1-space;

• is a T2-space is the set complement operator.

Closeness and separation

is closeThis 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.

See also

Notes

References

• .

.

• .

• .

• .

External links

• Alternative Characterizations of Topological Spaces