Kuratowski closure axioms

Remarks

Given a closure operator $c$, the sets $F$ with $c(F) = F$ are the closed sets of a unique topology on $X$. The closure of each $A$ in this topology is $c(A)$. Conversely, the closure in any topological space satisfies the four axioms. So closure operators on $X$ and topologies on $X$ correspond one to one.

Axiom 4 implies that $c$ is monotone: if $A \subseteq B$, then $c(A) \subseteq c(B) = c(A) \cup c(B \setminus A)$.

Example: on $\R$, let $c(A)$ be the set of limits of convergent sequences of points of $A$. This $c$ is the usual closure, and it satisfies the four axioms.

Non-example: fix a point $x_0$ of a nonempty set $X$ and let $c(A) = A \cup \set{x_0}$. Axioms 2, 3 and 4 hold, but $c(\emptyset) = \set{x_0} \ne \emptyset$, so axiom 1 fails.