Kuratowski closure axioms

Definition

Let X be a set and P ( X ) its power set.
A Kuratowski Closure Operator is an assignment cl : P ( X ) → P ( X ) with the following properties:

1. cl ⁡ ( ∅ ) = ∅ (Preservation of Nullary Union)
2. A ⊆ cl ⁡ ( A )  for every subset  A ⊆ X (Extensivity)
3. cl ⁡ ( A ∪ B ) = cl ⁡ ( A ) ∪ cl ⁡ ( B )  for any subsets  A , B ⊆ X (Preservation of Binary Union)
4. cl ⁡ ( cl ⁡ ( A ) ) = cl ⁡ ( A )  for every subset  A ⊆ X (Idempotence)

If the last axiom, idempotence, is omitted, then the axioms define a preclosure operator.

A consequence of the third axiom is: A ⊆ B ⇒ cl ⁡ ( A ) ⊆ cl ⁡ ( B ) (Preservation of Inclusion).

The four Kuratowski closure axioms can be replaced by a single condition, namely,

Induction of Topology

Construction
A closure operator naturally induces a topology as follows:
A subset C ⊆ X is called closed if and only if cl ⁡ ( C ) = C .

Empty Set and Entire Space are closed:
By extensitivity, X ⊆ cl ⁡ ( X ) and since closure maps the power set of X into itself (that is, the image of any subset is a subset of X ), cl ⁡ ( X ) ⊆ X we have X = cl ⁡ ( X ) . Thus X is closed.
The preservation of nullary unions states that cl ⁡ ( ∅ ) = ∅ . Thus ∅ is closed.

Arbitrary intersections of closed sets are closed:
Let I be an arbitrary set of indices and C i closed for every i ∈ I .
By extensitivity, ⋂ i ∈ I C i ⊆ cl ⁡ ( ⋂ i ∈ I C i ) .
Also, by preservation of inclusions, ⋂ i ∈ I C i ⊆ C i ∀ i ∈ I ⇒ cl ⁡ ( ⋂ i ∈ I C i ) ⊆ cl ⁡ ( C i ) = C i ∀ i ∈ I ⇒ cl ⁡ ( ⋂ i ∈ I C i ) ⊆ ⋂ i ∈ I C i .
Therefore, ⋂ i ∈ I C i = cl ⁡ ( ⋂ i ∈ I C i ) . Thus ⋂ i ∈ I C i is closed.

Finite unions of closed sets are closed:
Let I be a finite set of indices and let C i be closed for every i ∈ I .
From the preservation of binary unions and using induction we have ⋃ i ∈ I C i = cl ⁡ ( ⋃ i ∈ I C i ) . Thus ⋃ i ∈ I C i is closed.

Induction of closure

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: c l A ⁡ ( B ) = A ∩ c l X ⁡ ( B )  for all  B ⊆ A .

Recovering notions from topology

Closeness
A point p is close to a subset A iff p ∈ cl ⁡ ( A ) .

Continuity
A function f : X → Y is continuous at a point p iff p ∈ cl ⁡ ( A ) ⇒ f ( p ) ∈ cl ⁡ ( f ( A ) ) .