Finitely chainable and totally bounded metric spaces: Equivalent characterizations

Kundu, S.; Aggarwal, Manisha; Hazra, Somnath

doi:10.1016/j.topol.2016.11.008

Cited by 5 publications

(3 citation statements)

References 12 publications

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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: 99%

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: 99%

On Bourbaki-bounded sets on quasi-pseudometric spaces

Otafudu¹,

Mukonda²

2022

Math. Appl.

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“…Te notion of fnitely chainable metric space was introduced by Atsuji [13]. Kundu et al [14] collected equivalent conditions for fnite chainability in metric spaces. In 2002, Shrivastava and Agrawal [15] discovered the concept of ϵ-chainable sets in metric spaces.…”

Section: Introduction and Mathematical Preliminariesmentioning

confidence: 99%

Connected ϵ‐Chainable Sets and Existence Results

Bhandari¹,

Chandok

Jana³

et al. 2023

International Journal of Mathematics and Mathematical Sciences

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In the setting of ϵ -chainable metric spaces, we introduce ϵ − ρ − σ uniformly local weak contraction and obtain some results on the existence of fixed points. To show the veracity of the results, we also constructed some examples.

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“…Also, using these combinations of functions, we present some new characterizations of cofinally complete metric spaces and UC spaces. Here we should mention that the metric spaces on which every real-valued uniformly continuous function is bounded are precisely those which are finitely chainable, see [23]. On the other hand, the metric spaces on which every real-valued continuous function is bounded are precisely those which are compact.…”

Section: Introductionmentioning

confidence: 99%

Functions that preserve certain classes of sequences and locally Lipschitz functions

Gupta¹,

Kundu²

2020

Ann. Acad. Sci. Fenn. Math.

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The class of cofinally complete metric spaces lies between the class of complete metric spaces and that of compact metric spaces. It is known that a metric space (X, d) is cofinally complete if and only if every real-valued continuous function on (X, d) is cofinally Cauchy regular, where a function is said to be cofinally Cauchy regular or CC-regular for short if it preserves cofinally Cauchy sequences. Recently in 2017, Keremedis has defined almost bounded functions and AUC spaces [22]. We show that an AUC space is nothing but a cofinally complete metric space and an almost bounded function is nothing but a CC-regular function. Also in this paper, we study boundedness of various Lipschitz-type functions which are CC-regular as well and find equivalent characterizations of metric spaces on which such functions are uniformly continuous. Finally we explore some properties of cofinally Bourbaki-Cauchy regular functions, where a function is said to be cofinally Bourbaki-Cauchy regular if it preserves cofinally Bourbaki-Cauchy sequences [17] and find their relation with CC-regular functions.

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Finitely chainable and totally bounded metric spaces: Equivalent characterizations

Cited by 5 publications

References 12 publications

On Bourbaki-bounded sets on quasi-pseudometric spaces

On Bourbaki-bounded sets on quasi-pseudometric spaces

Connected ϵ‐Chainable Sets and Existence Results

Functions that preserve certain classes of sequences and locally Lipschitz functions

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