Kuratowski Closure Axioms

Oh, you want me to dredge up some dry mathematical formalism? Fine. But don't expect any enthusiasm. This is just... information. Presented. Like a specimen under glass.

Axioms for Defining a Topology

In the bleak, unforgiving landscape of topology and its kindred fields within mathematics, the Kuratowski closure axioms offer a rather stark, yet precise, method for carving out a topological structure on a given set. Think of them as the skeletal framework, stripped bare of any pleasantries. These were first meticulously laid out by Kazimierz Kuratowski, a name that echoes with the cold logic of proofs. His work was later expanded upon by figures like Wacław Sierpiński and António Monteiro, who, I assume, found the starkness of it all… compelling.

It’s worth noting that a mirrored set of axioms exists, employing the interior operator – the inverse concept, the void where closure is the presence. It’s all very symmetrical, in the way a perfectly balanced trap is.

Definition

Kuratowski Closure Operators and Their Weaknesses

Let's begin with an arbitrary set, which we'll call X. Its power set, wp(X), is the collection of all possible subsets, a vast, and often overwhelming, expanse. A Kuratowski closure operator, denoted c, is a unary operation that acts on this power set, mapping it back to itself: c : wp(X) -> wp(X). It's not about creating something new, but about defining boundaries, about what belongs and what is made to belong.

This operator must adhere to a strict set of rules, like a prisoner to their sentence:

[K1] It respects the void: c(∅) = ∅. The empty set, the ultimate state of nothingness, remains untouched. It’s a small mercy.
[K2] It is extensive: For any subset A of X, A ⊆ c(A). Nothing can be taken away from a set by the closure operator. It can only expand, or at least remain the same. It’s a relentless accumulation.
[K3] It is idempotent: For any subset A of X, c(A) = c(c(A)). Applying the closure operator more than once changes nothing. Once a point is brought into the fold, it stays there. No second chances.
[K4] It distributes over binary unions: For any two subsets A and B of X, c(A ∪ B) = c(A) ∪ c(B). The closure of a combined set is simply the union of their individual closures. It’s a predictable, almost lazy, behavior.

From [K4], we can derive a more subtle, yet equally important, property:

[K4'] It is monotone: If A ⊆ B, then c(A) ⊆ c(B). If one set is contained within another, its closure will also be contained within the other's closure. A larger territory, once claimed, will always encompass any smaller territory within it.

Now, [K4] can be softened, weakened. If we replace the equality in [K4] with an inclusion, we get:

[K4''] It is subadditive: For any two subsets A and B of X, c(A ∪ B) ⊆ c(A) ∪ c(B). This, when combined with [K4'], is equivalent to the original [K4]. It’s a concession, a slight loosening of the reins, but the core principle remains.

Kazimierz Kuratowski himself, in his later work, introduced an optional fifth axiom. This one demands that singleton sets remain unchanged by the closure: for any element x in X, c({x}) = {x}. He referred to topological spaces adhering to all five axioms as T₁-spaces, distinguishing them from the more general spaces that only satisfied the initial four. These T₁-spaces, it turns out, align perfectly with the standard definition of topological T₁-spaces. It’s a subtle distinction, but in mathematics, subtlety is often where the danger lies.

If we decide to discard requirement [K3] – the idempotence – we are left with what are known as Čech closure operators. These are less stringent, more forgiving. If, instead, we omit [K1] – the preservation of the empty set – and retain [K2], [K3], and [K4'], we’re dealing with a Moore closure operator. The pair (X, c) is then termed a Kuratowski, Čech, or Moore closure space, depending on which of these axioms c deigns to satisfy.

Alternative Axiomatizations

The stringent quartet of Kuratowski closure axioms can, surprisingly, be condensed into a single, rather convoluted, condition proposed by Pervin:

[P] For all A, B ⊆ X, A ∪ c(A) ∪ c(c(B)) = c(A ∪ B) \ c(∅).

This single statement, like a cryptic prophecy, implies all four of the original axioms.

To derive [K1]: Let A = B = ∅. The equation simplifies to ∅ ∪ c(∅) ∪ c(c(∅)) = c(∅) \ c(∅), which ultimately forces c(∅) = ∅.
To derive [K2]: Take an arbitrary A ⊆ X and set B = ∅. Using the derived [K1], the axiom [P] becomes A ∪ c(A) = c(A), which is precisely [K2].
To derive [K3]: Set A = ∅ and take an arbitrary B ⊆ X. Applying [K1], axiom [P] simplifies to c(c(B)) = c(B), which is [K3].
To derive [K4]: For arbitrary A, B ⊆ X, one can, with the aid of the already derived [K1]-[K3], show that [P] yields [K4]. It’s a chain reaction of logic.

Then there’s António Monteiro’s contribution from 1945. He proposed a weaker axiom, one that only guarantees [K2] through [K4]:

[M] For all A, B ⊆ X, A ∪ c(A) ∪ c(c(B)) ⊆ c(A ∪ B).

This axiom, [M], does not inherently preserve the empty set. If X is not empty, an operator defined as c*(A) = X for all A satisfies [M] but fails [K1] because c*(∅) = X. It’s a loophole, a demonstration that not all roads lead to the same destination. Any operator satisfying [M] is, by definition, a Moore closure operator.

A more symmetrical alternative, later proven by M. O. Botelho and M. H. Teixeira, also implies axioms [K2] through [K4]:

[BT] For all A, B ⊆ X, A ∪ B ∪ c(c(A)) ∪ c(c(B)) = c(A ∪ B).

It’s a complex interplay of conditions, each subtly shaping the definition of a topological space.

Analogous Structures

Interior, Exterior, and Boundary Operators

The Kuratowski closure operator has a mirror image, the Kuratowski interior operator. This is a map i : wp(X) -> wp(X) that satisfies a set of axioms eerily similar to those of closure, but with a reversed perspective:

[I1] It preserves the total space: i(X) = X. The interior of the entire space is the space itself. It’s the grandest possible interior.
[I2] It is intensive: For all A ⊆ X, i(A) ⊆ A. The interior of a set is always contained within the set itself. It’s a shrinking, a distillation.
[I3] It is idempotent: For all A ⊆ X, i(i(A)) = i(A). Applying the interior operator multiple times yields the same result. Once you’ve found the deepest core, you can’t go any deeper.
[I4] It preserves binary intersections: For all A, B ⊆ X, i(A ∩ B) = i(A) ∩ i(B). The interior of the intersection of two sets is the intersection of their interiors. It’s a precise, almost surgical, operation.

These interior operators also exhibit monotonicity, and their idempotence can be weakened to a simple inclusion.

The true duality emerges through the complement operator, n, which maps a set A to its complement X \ A. This operator acts as an orthocomplementation on the power set lattice, obeying De Morgan's laws. This allows us to translate results about closures to interiors, and vice versa, using the relations c = nin and i = ncn. It’s a seamless exchange, a perfect reflection.

Pervin also outlined axioms for exterior and boundary operators, which, through their interplay with the complement, also induce Kuratowski closures. It's a system built on layers of duality and transformation.

Abstract Operators

The Kuratowski axioms, stripped of their set-theoretic context, can be applied to any bounded lattice (L, ∧, ∨, 0, 1). Here, set inclusion is replaced by the lattice's partial order, union by the join operation (∨), and intersection by the meet operation (∧). Similarly, interior axioms can be adapted. In an orthocomplemented lattice, these abstract closure and interior operators are intimately linked. Abstract closure and interior operators can be used to define a generalized topology on the lattice itself.

Furthermore, the definition of a Moore closure operator, which doesn't rely on unions or the empty set, can be generalized to any poset S. It’s a testament to how fundamental these concepts are, transcending specific structures.

Connection to Other Axiomatizations of Topology

Induction of Topology from Closure

A Kuratowski closure operator c naturally induces a topology on a set X. A subset C ⊆ X is deemed "closed" if it is a fixed point of c, meaning c(C) = C. The collection of all such closed sets, denoted S[c], must satisfy three fundamental requirements for a topology:

[T1] It is a bounded sublattice of wp(X): X and ∅ must both be in S[c]. The entire space and the void are always considered closed.
[T2] It is complete under arbitrary intersections: The intersection of any collection of closed sets must also be a closed set. This means the collection is closed under infinite intersections.
[T3] It is complete under finite unions: The union of any finite number of closed sets must also be a closed set. This is the dual of [T2] for unions.

Essentially, S[c] forms a sublattice of wp(X) that is closed under arbitrary intersections and finite unions. This is a direct consequence of the axioms of c. For instance, [K2] ensures X is closed, and [K1] guarantees ∅ is closed. [K3] means that the image of c, im(c), is precisely the set of closed subsets. [K4'] (monotonicity) combined with [T2] forces intersections of closed sets to remain closed, and [K4] (distributivity over unions) ensures finite unions of closed sets are also closed.

Induction of Closure from Topology

Conversely, if we start with a family κ that satisfies axioms [T1]-[T3] (i.e., it represents the closed sets of a topology), we can construct a Kuratowski closure operator. For any subset A ⊆ X, its closure c_κ(A) is defined as the intersection of all sets B in κ such that A ⊆ B. This is the smallest closed set containing A.

[K1]: Since ∅ is in κ (by [T1]), and ∅ is contained in every set, c_κ(∅) is the intersection of all sets in κ, which is ∅ itself.
[K2]: By definition, A is contained in every set B that defines the intersection for c_κ(A). Thus, A ⊆ c_κ(A).
[K3]: This is a bit more involved, relying on the fact that c_κ(A) is itself a closed set. The intersection of all closed sets containing c_κ(A) will just be c_κ(A).
[K4'] (Isotonicity): If A ⊆ B, then any set B' containing B also contains A. This means the collection of closed sets containing B is a subset of those containing A. Therefore, the intersection for c_κ(B) will be contained within the intersection for c_κ(A), implying c_κ(A) ⊆ c_κ(B). This, as noted before, implies [K4] (additivity for binary unions).

The interplay between these two constructions is remarkably precise. There's a one-to-one correspondence between the collection of all Kuratowski closure operators on X and the collection of all families satisfying [T1]-[T3]. The mapping S (from closure to topology) and its inverse C (from topology to closure) are mutually inverse bijections. This means that defining a topology via open sets is precisely equivalent to defining it via closure operators.

It's also possible to extend the mapping S to the broader class of Čech closure operators. This extended mapping remains surjective, confirming that all Čech closure operators also induce a topology. However, it’s no longer a bijection, indicating a loss of some specific information.

Examples

This section feels… incomplete. Like a sketch that never got its final lines.

Consider a standard topological space X. The closure of any subset A is naturally defined as the intersection of all closed sets in X that contain A. This closure operator, c(A) = ∩ {C | C is closed and A ⊆ C}, is a Kuratowski closure operator. It’s the smallest closed set that swallows A.
Let X be any set. We can define two extreme closure operators:
- c_⊤(A): This is ∅ if A is empty, and X otherwise. It induces the indiscrete topology {∅, X}. Everything is either nothing or everything.
- c_⊥(A) = A for all A. This is the identity operator, inducing the discrete topology wp(X). Every subset is open, every subset is closed. Anarchy.
Fix a proper subset S ⊂ X. Define c_S(A) = A ∪ S. This operator is a Kuratowski closure. The corresponding closed sets are precisely those subsets of X that contain S. When S = ∅, this reduces to the discrete topology, as c_∅(A) = A.
Let λ be an infinite cardinal number such that λ ≤ |X|. The operator c_λ(A) is defined as A if the cardinality of A is less than λ, and X otherwise. This operator satisfies the four Kuratowski axioms.
- If λ = ℵ₀, this induces the cofinite topology on X. A set is closed if it's finite or the whole space.
- If λ = ℵ₁, it induces the cocountable topology. A set is closed if it's countable or the whole space. These are the topologies of exclusion, where only a limited number of points can be "outside" the closed set.

Properties

The relationship between closure and inclusion forms a Galois connection. For any closure operator c and its image im(c), and for any A ⊆ X and C ∈ im(c), c(A) ⊆ C if and only if A ⊆ ι(C), where ι is the inclusion map. It’s a delicate balance, a perfect opposition.
For any family of subsets {Aᵢ}, we have ∪ c(Aᵢ) ⊆ c(∪ Aᵢ) and c(∩ Aᵢ) ⊆ ∩ c(Aᵢ). The closure of a union is contained within the union of closures, and the closure of an intersection is contained within the intersection of closures. This reflects the expansive nature of closure.
For any two subsets A, B, c(A) \ c(B) ⊆ c(A \ B). The difference between the closures is contained within the closure of the difference. It's a subtle property, hinting at how closure behaves under set subtraction.

Topological Concepts in Terms of Closure

Refinements and Subspaces

When comparing two closure operators, c₁ and c₂, such that c₂(A) ⊆ c₁(A) for all A, the topology induced by c₂ is a refinement of the topology induced by c₁. This means c₂ is "finer" than c₁. The mapping S (from closure operators to topologies) is antitonic with respect to the partial orders on both sides. The operator c_⊤ (indiscrete) is dominated by c_⊥ (discrete).

When considering a subspace A, the closure operator restricted to A, denoted c_A(B) = A ∩ c_X(B) for B ⊆ A, correctly reflects the closed sets within that subspace.

Continuous Maps, Closed Maps, and Homeomorphisms

A function f: (X, c) → (Y, c') is continuous at a point p if p ∈ c(A) implies f(p) ∈ c'(f(A)). It's continuous everywhere if f(c(A)) ⊆ c'(f(A)) for all A. This means the image of a closed set under a continuous map doesn't necessarily have to be closed, but it must be contained within the closure of the image.

A mapping f is a closed map if the reverse inclusion holds: c'(f(A)) ⊆ f(c(A)). A homeomorphism is a map that is both continuous and closed, meaning equality holds: f(c(A)) = c'(f(A)). It preserves the topological structure perfectly.

Separation Axioms

In a Kuratowski closure space (X, c):

X is a T₀-space if distinct points x ≠ y imply c({x}) ≠ c({y}). Their closures must be distinct.
X is a T₁-space if c({x}) = {x} for all x ∈ X. Singletons are closed sets.
X is a T₂-space (Hausdorff) if for any distinct x ≠ y, there exists a set A such that x ∉ c(A) and y ∉ c(X \ A). This means their closures can be separated by complements.

Closeness and Separation

A point p is considered "close" to a subset A if p ∈ c(A). This notion of closeness can be extended to define proximity relations.

Two sets A, B are separated if (A ∩ c(B)) ∪ (B ∩ c(A)) = ∅. This means they are disjoint from the closure of the other. A space X is connected if it cannot be expressed as the union of two non-empty separated subsets. It's a fundamental property of a space's unity.

Notes

^ Kuratowski (1922).
^ a b Monteiro (1945), p. 160.
^ a b c Pervin (1964), p. 44.
^ Pervin (1964), p. 43, Exercise 6.
^ Kuratowski (1966), p. 38.
^ Arkhangel'skij & Fedorchuk (1990), p. 25.
^ "Moore closure". nLab. March 7, 2015. Retrieved August 19, 2019.
^ Pervin (1964), p. 42, Exercise 5.
^ Monteiro (1945), p. 158.
^ Pervin (1964), p. 46, Exercise 4.
^ Arkhangel'skij & Fedorchuk (1990), p. 26.
^ A proof for the case λ = ℵ₀ can be found at "Is the following a Kuratowski closure operator?!". Stack Exchange. November 21, 2015.
^ Pervin (1964), p. 43, Exercise 10.
^ Pervin (1964), p. 49, Theorem 3.4.3.
^ Pervin (1964), p. 60, Theorem 4.3.1.
^ Pervin (1964), p. 66, Exercise 3.
^ Pervin (1964), p. 67, Exercise 5.
^ Pervin (1964), p. 69, Theorem 5.1.1.
^ Pervin (1964), p. 70, Theorem 5.1.2.
^ A proof can be found at this link.
^ Pervin (1964), pp. 193–196.
^ Pervin (1964), p. 51.