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.