RT Journal Article
T1 The Samuel realcompactification of a metric space
A1 Garrido Carballo, María Isabel
A1 Meroño Moreno, Ana Soledad
AB In this paper we introduce a realcompactification for any metric space (X, d), defined by means of the family of all its real-valued uniformly continuous functions. We call it the Samuel realcompactification, according to the well known Samuel compactification associated to the family of all the bounded real-valued uniformly continuous functions. Among many other things, we study the corresponding problem of the Samuel realcompactness for metric spaces. At this respect, we prove that a result of Katětov–Shirota type occurs in this context, where the completeness property is replaced by Bourbaki-completeness (a notion recently introduced by the authors) and the closed discrete subspaces are replaced by the uniformly discrete ones. More precisely, we see that a metric space (X, d) is Samuel realcompact iff it is Bourbaki-complete and every uniformly discrete subspace of X has non-measurable cardinal. As a consequence, we derive that a normed space is Samuel realcompact iff it has finite dimension. And this means in particular that realcompactness and Samuel realcompactness can be very far apart. The paper also contains results relating this realcompactification with the so-called Lipschitz realcompactification (also studied here), with the classical Hewitt–Nachbin realcompactification and with the completion of the initial metric space
PB Elsevier
SN 0022-247X
YR 2017
FD 2017
LK https://hdl.handle.net/20.500.14352/100602
UL https://hdl.handle.net/20.500.14352/100602
LA eng
NO Garrido MI, Meroño AS. The Samuel realcompactification of a metric space. Journal of Mathematical Analysis and Applications 2017;456:1013–39. https://doi.org/10.1016/j.jmaa.2017.07.033.
NO Ministerio de Economía, Comercio y Empresa (España)
DS Docta Complutense
RD 6 abr 2025