Two classes of metric spaces

Garrido, M. Isabel; Meroño, Ana S.

doi:10.4995/agt.2016.4401

Cited by 4 publications

(5 citation statements)

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“…The first two inclusions of (4.3) can be found in [2] and [5], for example, and the last inclusion of (4.3) is a consequence of Proposition 4.12. For more details about connections between bornology of q s -totally bounded sets, bornology of q s -Bourbaki-bounded sets and bornology of q s -bounded sets, we recommend, for instance, [2,5,7,8].…”

Section: Some First Resultsmentioning

confidence: 94%

“…The theory of Bourbaki-boundedness in metric spaces was introduced by Atsuji in [1] as a generalization of the concept of totally bounded metric spaces. However, the concept of Bourbaki-boundedness attracted a great interest of many scholars (see [5][6][7]). For instance in [6], the authors introduced new tools for the completeness of metric spaces, called Bourbaki-completeness and cofinal Bourbaki completeness.…”

Section: Introductionmentioning

confidence: 99%

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On Bourbaki-bounded sets on quasi-pseudometric spaces

Otafudu¹,

Mukonda²

2022

Math. Appl.

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In metric spaces, a set is Bourbaki-bounded if and only if every realvalued uniformily continuous function on it is bounded. In this article, we study Bourbaki-boundedness on quasi-pseudometric spaces. It turns out that if a set is Bourbaki-bounded on a symmetrized quasi-pseudometric space, then it is Bourbakibounded in the quasi-metric space but the converse need not to be true. We show that an asymmetric normed space is Bourbaki-bounded if and only if it is bounded. Consequently, we prove that every real-valued semi-Lipschitz in the small function on a quasi-metric space is bounded if and only if the quasi-metric is Bourbaki-bounded. This article extends some results from Beer and Garrido's paper [2] from the metric point of view to the context of quasi-metric spaces.

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Section: Some First Resultsmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

On Bourbaki-bounded sets on quasi-pseudometric spaces

Otafudu¹,

Mukonda²

2022

Math. Appl.

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“…For instance, if we take the set of natural numbers N endowed with the usual metric, then the space N × ℓ 2 satisfies that every bounded set is Bourbaki-bounded, but it is not small-determined neither it satisfies the Heine-Borel property. We refer to the paper [15] where we see that spaces for which BB d (X) = B d (X) have the property that every uniform partition is in fact countable, and also that they are properly located between two known classes of metric spaces, namely the above mentioned small-determined spaces and the so-called B-simple spaces introduced by Hecjman in [19].…”

Section: Some Results Related To Bornologiesmentioning

confidence: 99%

“…In the general context of uniform spaces we refer to the paper by Hejcman [18] where they are called uniformly bounded subsets. Very recently Bourbaki-boundedness have been considered (with this precise name) in different frameworks, see for instance [6], [7], [13], [14] and [15].…”

Section: Some Types Of Uniform Boundednessmentioning

confidence: 99%

The Samuel realcompactification of a metric space

Garrido

Meroño

2017

Journal of Mathematical Analysis and Applications

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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 Kat\v{e}tov-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

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“…Garrido and Merono [20] characterized Bourbaki boundedness in terms of new sequences named as Bourbaki Cauchy and cofinally Bourbaki Cauchy. For more results related to these new concepts, one can see [21,22]. By using these new general Cauchy sequences, the authors defined new types of completeness for metric spaces called as Bourbaki complete and cofinally Bourbaki complete.…”

Section: Introductionmentioning

confidence: 99%

Bourbaki completeness in quasi metric spaces

2019

J. Nonlinear Funct. Anal.

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Two classes of metric spaces

Cited by 4 publications

References 5 publications

On Bourbaki-bounded sets on quasi-pseudometric spaces

On Bourbaki-bounded sets on quasi-pseudometric spaces

The Samuel realcompactification of a metric space

Bourbaki completeness in quasi metric spaces

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