Realcompactness and spaces of vector-valued functions
Tom 172 / 2002
Fundamenta Mathematicae 172 (2002), 27-40
MSC: Primary 54C35; Secondary 54C40, 54D60, 46E40.
DOI: 10.4064/fm172-1-3
Streszczenie
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.
