A closure operator on a set is an operator which assigns a set to each subset of , and such that the following (Kuratowski's closure axioms) hold for any subsets and of :

1. ;
2. ;
3. ;
4. 

The following theorem due to Kuratowski says that a closure operator characterizes a unique topology on :

Theorem. Let be a closure operator on , and let . Then is a topology on , and is the -closure of for each subset of .