The Kuratowski closure axioms allow a topology to be defined on a set in terms of a closure operator which assigns each subset its closure.

The axioms of a closure operator are:

1. (i) Every subset is contained in its closure.

2. (ii) The closure of the closure of a subset equals the closure of the subset.

3. (iii) The closure of the union of two subsets is the union of their closures.

4. (iv) The closure of the empty set is empty.

Given a closure operator, closed subsets can be defined as those sets which equal their closure, and the complements of the closed subsets, i.e. the open subsets, satisfy the axioms of being a topology.