Axiomatization of a Topology using the Boundary
Solution 1:
As it known, $\partial A=c(A)\cap c(X\setminus A)$. Therefore, since $c(A)\supset A$, you can restore closure operator (and then topology) by the given boundary operator, when you let $c(A):=A\cup \partial A$.
For the boundary correctly determine the closure, following axioms have to hold:
$\partial A=\partial(X\setminus A)$ for any $A$;$\,\,\,$ [for satisfy redefinition]
$\partial \emptyset=\emptyset$;$\,\,\,$ [for satisfy KC1]
$\partial(\partial A)\subset\partial A$ for any $A$;$\,\,\,$ [for satisfy KC3]
$\partial(A\cup B)\subset \partial A\cup\partial B$ for any $A,B$;$\,\,\,$ [for satisfy KC4]
and KC2 will be satisfied automatically.
EDIT: for really satisfy CK4, besides axiom 4 we need also
- $A\subset B\Rightarrow \partial A\subset B\cup\partial B$.