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

1. $\emptyset^c = \emptyset$ ;
2. $A\subset A^c$ ;
3. $(A^c)^c = A^c$ ;
4. $(A\cup B)^c = A^c\cup B^c.$

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

Theorem. Let $c$ be a closure operator on $X$ , and let $\mathcal{T} = \{X-A: A\subseteq X,\; A^c=A\}$ . Then $\mathcal{T}$ is a topology on $X$ , and $A^c$ is the $\mathcal{T}$ -closure of $A$ for each subset $A$ of $X$ .