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 interior operator.

Let ${\displaystyle X}$ be an arbitrary set and ${\displaystyle \wp (X)}$ its power set. A Kuratowski closure operator is a unary operation ${\displaystyle \mathbf {c} :\wp (X)\to \wp (X)}$ with the following properties:

[K2] It is extensive: for all ${\displaystyle A\subseteq X}$, ${\displaystyle A\subseteq \mathbf {c} (A)}$;

[K3] It is idempotent: for all ${\displaystyle A\subseteq X}$, ${\displaystyle \mathbf {c} (A)=\mathbf {c} (\mathbf {c} (A))}$;

A consequence of ${\displaystyle \mathbf {c} }$ preserving binary unions is the following condition:

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).