For a commutative ring $A$ you can define the integral closure of $A$ as $$\overline{A}^{\operatorname{int}}:=\lbrace x\in \operatorname{Quot}(A)\mid x\text{ is integral over } A \rbrace.$$ Since this notation resembles the topological closure of a set, I wondered if there are any topologies (of number theoretical interest) on the algebraic closure $K$ of $A$, in which the integral closures $\overline{B}$ of all rings $A\subseteq B\subseteq K$ are closed sets.
Is anything known about that kind of topologies? Are there interesting examples?
In particular: Which topological properties does the topology on $K$ which is generated by the complements of $\left\lbrace \overline{B}^{\operatorname{int}} \mid B \text{ is a ring and } A\subseteq B\subseteq K \right\rbrace$ have?
Every closure operator induces a topology, and every topology is induced by a closure operator. That is, there is a bijection between closure spaces, that is, spaces where a closure operator has been defined and where closed sets are the fixed points of this operator, and topological spaces.