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 introduced by Kazimierz Kuratowski.
A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator.
Let be a set and its power set.
A Kuratowski Closure Operator is an assignment with the following properties:
- (Preservation of Nullary Union)
- (Preservation of Binary Union)
If the last axiom, idempotence, is omitted, then the axioms define a preclosure operator.
A consequence of the third axiom is: (Preservation of Inclusion).
The four Kuratowski closure axioms can be replaced by a single condition, namely,
Connection to other axiomatizations of topology
Induction of Topology
A closure operator naturally induces a topology as follows:
A subset is called closed if and only if .
Empty Set and Entire Space are closed:
By extensitivity, and since closure maps the power set of into itself (that is, the image of any subset is a subset of ), we have . Thus is closed.
The preservation of nullary unions states that . Thus is closed.
Arbitrary intersections of closed sets are closed:
Let be an arbitrary set of indices and closed for every .
Also, by preservation of inclusions,
Therefore, . Thus is closed.
Finite unions of closed sets are closed:
Let be a finite set of indices and let be closed for every .
From the preservation of binary unions and using induction we have . Thus 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: 
Recovering notions from topology
A point is close to a subset iff .
A function is continuous at a point iff .