In some cases, a topological space is completely characterized by (not necessarily topological) properties of associated spaces of continuous functions. For instance, if X and Y are realcompact, an isomorphism between the rings C(X) and C(Y) leads to a homeomorphism between X and Y. Moreover, if X and Y are compact, then the same conclusion is reached if there exists a linear isometry between the Banach spaces C(X) and C(Y).
In this talk we will modify the ring isomorphism condition obtaining an analogous result. We will introduce separating (or disjointness preserving) maps. This will allow us to study a larger family of spaces, particularly vector-valued. In these cases, we can get a description of these maps, and as a result study their continuity. As an application, we will see that isometries between spaces of vector-valued functions defined on noncompact (and not locally compact) spaces have also a special form.