Some topological considerations on semistar operations

We consider properties and applications of a new topology, called the Zariski topology  , on the space SStar(A)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 SStar(A)SStar(A). The topology on SStar(A)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 SStarf(A)SStarf(A) of all the semistar operations of finite type on A is a spectral space.