closure axioms

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. ${\mathrm{\varnothing }}^c=\mathrm{\varnothing }$;

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 $𝒯=\{X-A:A\subseteq X,A^c=A\}$. Then $𝒯$ is a topology on $X$, and $A^c$ is the $𝒯$-closure of $A$ for each subset $A$ of $X$.