2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/60213It is shown that the existence of a biseparating map between a large class of spaces of vector-valued continuous functions A(X,E) and A(Y,F) implies that some compactifications of X and Y are homeomorphic. In some cases, conditions are given to warrant the existence of a homeomorphism between the realcompactifications of X and Y; in particular we find remarkable differences with respect to the scalar context: namely, if E and F are infinite-dimensional and T is a biseparating map between the space of E-valued bounded continuous functions on X and that of F-valued bounded continuous functions on Y, then the realcompactifications of X and Y are homeomorphic.15 pages, LaTeX. Results stated for arbitrary normed spaces without changes in proofs. New presentation and new examples. One reference addedGeneral TopologyFunctional Analysis54C35 (Primary) 54C40, 54D60, 46E40 (Secondary)text