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Garrido, M. Isabel and Meroño, Ana S. (2017) The Samuel realcompactification of a metric space. Journal of Mathematical Analysis and Applications, 456 (2). pp. 10131039. ISSN 0022247X

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Official URL: http://www.sciencedirect.com/science/article/pii/S0022247X17306960
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Abstract
In this paper we introduce a realcompactification for any metric space (X, d), defined by means of the family of all its realvalued uniformly continuous functions. We call it the Samuel realcompactification, according to the well known Samuel compactification associated to the family of all the bounded realvalued 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 KatetovShirota type occurs in this context, where the completeness property is replaced by Bourbakicompleteness (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 Bourbakicomplete and every uniformly discrete subspace of X has nonmeasurable 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 socalled Lipschitz realcompactification (also studied here), with the classical HewittNachbin realcompactification and with the completion of the initial metric space.
Item Type:  Article 

Uncontrolled Keywords:  Realvalued uniformly continuous functions; Samuel realcompactification; Lipschitz realcompactification; Bourbakiboundedness; Bourbakicompleteness; Uniform KatetovShirota result 
Subjects:  Sciences > Mathematics > Differential geometry 
ID Code:  45124 
Deposited On:  18 Oct 2017 09:42 
Last Modified:  12 Dec 2018 15:12 
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