Realcompactness and spaces of vector-valued functions

Volume 172 / 2002

Jesus Araujo Fundamenta Mathematicae 172 (2002), 27-40 MSC: Primary 54C35; Secondary 54C40, 54D60, 46E40. DOI: 10.4064/fm172-1-3


It 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 $T: C^{*} (X,E) \rightarrow C^{*} (Y, F)$ is a biseparating map, then the realcompactifications of $X$ and $Y$ are homeomorphic.


  • Jesus AraujoDepartamento de Matemáticas, Estadística y Computación
    Facultad de Ciencias
    Universidad de Cantabria
    Avenida de los Castros s.n.
    E-39071 Santander, Spain

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