Some topological considerations on semistar operations

We consider properties and applications of a new topology, called \emph{the Zariski topology}, on the space $\mathrm{SStar}(A)$ of all the semistar operations on an integral domain $A$. We prove that the set of all overrings of $A$, endowed with the classical Zariski topology, is homeomorphic to a subspace of $\mathrm{SStar}(A)$. The topology on $\mathrm{SStar}(A)$ provides a general theory, through which we see several algebraic properties of semistar operation as very particular cases of our construction. Moreover, we show that the subspace $\mathrm{SStar}_f(A)$ of all the semistar operations of finite type on $A$ is a spectral space.