Topological and geometrical properties of spaces with symmetric and nonsymmetric f-quasimetrics

Arutyunov, A. V.; Грешнов, А. В.; Локуциевский, Лев Вячеславович; Storozhuk, K. V.

doi:10.1016/j.topol.2017.02.035

Cited by 26 publications

(13 citation statements)

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“…The next theorem provides such conditions. However, it is worth noting here that a similar result appeared in [3] (see [3,Proposition 4.1]), where the authors proved the implication (i) =⇒ (ii) formulated below. Here we expand the list with two additional conditions.…”

Section: The Second Refinementsupporting

confidence: 70%

“…Topology in semimetric spaces can be defined in various ways. The first and most common approach is to define a topology in the following way: [2,3,11,25]. What makes this topology particularly useful is the fact that the convergence of a sequence (x n ) ∈ X N to some point x ∈ X with respect to this topology is equivalently described by d(x n , x) → 0.…”

Section: Definition 22mentioning

confidence: 99%

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Two refinements of Frink’s metrization theorem and fixed point results for Lipschitzian mappings on quasimetric spaces

2018

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Quasimetric spaces have been an object of thorough investigation since Frink's paper appeared in 1937 and various generalisations of the axioms of metric spaces are now experiencing their well-deserved renaissance. The aim of this paper is to present two improvements of Frink's metrization theorem along with some fixed point results for single-valued mappings on quasimetric spaces. Moreover, Cantor's intersection theorem for sequences of sets which are not necessarily closed is established in a quasimetric setting. This enables us to give a new proof of a quasimetric version of the Banach Contraction Principle obtained by Bakhtin. We also provide error estimates for a sequence of iterates of a mapping, which seem to be new even in a metric setting.

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Section: The Second Refinementsupporting

confidence: 70%

Section: Definition 22mentioning

confidence: 99%

Two refinements of Frink’s metrization theorem and fixed point results for Lipschitzian mappings on quasimetric spaces

2018

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“…A particular case of the next example can be found in [3]. Here the space (X, d s ) is not compact, whereas the space (X, inf d s ) is compact.…”

Section: If G ⊂ (X ρ) Is a Curve With Endpoints A B Satisfying The mentioning

confidence: 99%

Metrics 𝜌, quasimetrics 𝜌^{𝑠} and pseudometrics inf𝜌^{𝑠}

Storozhuk

2017

Conform. Geom. Dyn.

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Abstract. Let ρ be a metric on a space X and let s ≥ 1. The function ρ s (a, b) = ρ(a, b) s is a quasimetric (it need not satisfy the triangle inequality). The function inf ρ s (a, b) defined by the condition inf ρ s (a, b) = inf{ n 0 ρ s (z i , z i+1 ) z 0 = a, z n = b} is a pseudometric (i.e., satisfies the triangle inequality but can be degenerate). We show how this degeneracy can be connected with the Hausdorff dimension of the space (X, ρ). We also give some examples showing how the topology of the space (X, inf ρ s ) can change as s changes.

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“…Настоящая работа основана на подходах и технике, развитой в [3,4,5]. Также следует отметить работу [7], в которой была изучена топология f -квазиметрических пространтв, включающих в себя и (q 1 , q 2 )квазиметрические пространства; в [7], в частности, читатель может найти для себя необходимые разъяснения и иллюстрации тех или иных топологических объектов f -квазиметрических пространств. П.…”

Section: Introductionunclassified