Kuratowski closure axioms

In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which 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 introduced by Kazimierz Kuratowski, in a slightly different form that applied only to Hausdorff spaces.

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

Definition

A topological space $(X,\operatorname{cl})$ is a set X with a function

$\operatorname{cl}:\mathcal{P}(X) \to \mathcal{P}(X)$

called the closure operator where $\mathcal{P}(X)$ is the power set of X.

The closure operator has to satisfy the following properties for all $A, B\in\mathcal{P}(X)$

1. $A \subseteq \operatorname{cl}(A) \!$ (Extensivity)
2. $\operatorname{cl}(\operatorname{cl}(A)) = \operatorname{cl}(A) \!$ (Idempotence)
3. $\operatorname{cl}(A \cup B) = \operatorname{cl}(A) \cup \operatorname{cl}(B) \!$ (Preservation of binary unions)
4. $\operatorname{cl}(\varnothing) = \varnothing \!$ (Preservation of nullary unions)

If the second axiom, that of idempotence, is relaxed, then the axioms define a preclosure operator.

Notes

By induction, Axioms (3) and (4) are equivalent to the single statement

$\operatorname{cl}(A_{1} \cup \cdots \cup A_{n}) = \operatorname{cl}(A_{1}) \cup \cdots \cup \operatorname{cl}(A_{n}), n \geq 0 \!$ (Preservation of finitary unions).

Recovering topological definitions

A function between two topological spaces

$f:(X,\operatorname{cl}) \to (X',\operatorname{cl}')$

is called continuous if for all subsets A of X

$f(\operatorname{cl}(A)) \subset \operatorname{cl}'(f(A))$

A point p is called close to A in $(X,\operatorname{cl})$ if $p\in \operatorname{cl}(A)$

A is called closed in $(X,\operatorname{cl})$ if $A=\operatorname{cl}(A)$. In other words the closed sets of X are the fixed points of the closure operator.

If one takes an "open set" to be a set whose complement is closed, then the family of all open sets forms a topology. Conversely, any topology can be induced in this way by the correct choice of closure operator.

See also

• Preclosure operator