Some topological considerations on semistar operations

We consider properties and applications of a new topology, called the Zariski topology, on the space $\mathrm{S}\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{r}(A)$ 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 ${\rm SStar}(A)$\mathrm{.}Thetopologyon${\rm SStar}(A)$ 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 ${\rm SStar}_f(A)$ofallthesemistaroperationsoffinitetypeon$A$ is a spectral space.