Large Scale Versus Small Scale

Cencelj, Matija; Dydak, Jerzy; Vavpetič, Aleš

doi:10.2991/978-94-6239-024-9_4

Cited by 5 publications

(13 citation statements)

References 42 publications

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“…It is shown in [11] (see Theorem 9.1 (5)) that a topological space X is collectionwise normal if and only if partitions of unity on each closed subset A of X extends over X. In other words, certain spaces are absolute extensors of X.…”

Section: Coarse Normalitymentioning

confidence: 99%

“…Definition 3.1 [2,3]. A metric space X is large-scale weakly paracompact if for each r, s > 0 there is a uniformly bounded cover U of X of Lebesgue number at least s such that every r-ball B(x, r) is contained in only finitely many elements of U.…”

Section: Large-scale Weak Paracompactnessmentioning

confidence: 99%

“…Subsequently, it was generalized to the concept of exact spaces by Dadarlat and Guentner [7]. In [4] exact spaces were narrowed down to large scale paracompact spaces and [6] (see also [5]) contains an analysis of interrelationships between various concepts.…”

Section: Introductionmentioning

confidence: 99%

“…One may summarize that the three concepts (Property A, exact spaces, and largescale paracompact spaces) deal with dualizing paracompactness via partitions of unity. c 2015 Australian Mathematical Publishing Association Inc. 1446-7887/2015 $16.00 J. Dydak [2] In [3], the concept of Strong Property A was introduced as a way of dualizing paracompactness via covers. This paper is devoted to developing large-scale paracompactness from the point of view of discreteness.…”

Section: Introductionmentioning

confidence: 99%

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Coarse Amenability and Discreteness

Dydak¹

2015

J. Aust. Math. Soc.

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This paper is devoted to dualization of paracompactness to the coarse category via the concept of $R$-disjointness. Property A of Yu can be seen as a coarse variant of amenability via partitions of unity and leads to a dualization of paracompactness via partitions of unity. On the other hand, finite decomposition complexity of Guentner, Tessera, and Yu and straight finite decomposition complexity of Dranishnikov and Zarichnyi employ $R$-disjointness as the main concept. We generalize both concepts to that of countable asymptotic dimension and our main result shows that it is a subclass of spaces with Property A. In addition, it gives a necessary and sufficient condition for spaces of countable asymptotic dimension to be of finite asymptotic dimension.

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Section: Coarse Normalitymentioning

confidence: 99%

Section: Large-scale Weak Paracompactnessmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Coarse Amenability and Discreteness

Dydak¹

2015

J. Aust. Math. Soc.

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“…That means there is a real number S > 0 with each member of U i being of diameter at most S and the distance between points belonging to different elements of U i is at least r.Since then several concepts related to asymptotic dimension were introduced by various authors. One can see them as a spectrum with asymptotic dimension being the strongest concept and weak coarse paracompactness being the weakest (see [7]). The concept closest to asymptotic dimension was introduced by Dranishnikov [9] under the name of Asymptotic Property C: Definition 1.2.…”

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confidence: 99%

Matrix algebra of sets and variants of decomposition complexity

Dydak

2019

Rev Mat Complut

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We introduce matrix algebra of subsets in metric spaces and we apply it to improve results of Yamauchi and Davila regarding Asymptotic Property C. Here is a representative result: Suppose X is an ∞-pseudo-metric space and n ≥ 0 is an integer. The asymptotic dimension asdim(X) of X is at most n if and only if for any real number r > 0 and any integer m ≥ 1 there is an augmented m× (n + 1)-matrix M = [B|A] (that means B is a column-matrix and A is an m × n-matrix) of subspaces of X of scale-r-dimension 0 such that M ·∩ M T is bigger than or equal to the identity matrix and B(A, r) ·∩ B(A, r) T is a diagonal matrix. IntroductionThis paper is devoted to decomposition complexity understood as any coarse invariant defined in terms of decomposing spaces into r-disjoint subsets. Historically, the first such invariant, namely asymptotic dimension, was introduced by Gromov for the purpose of studying groups using geometric methods (see [32]). In Ostrand ([26] or [27]) formulation (see [4]) it can be defined as follows:Definition 1.1. Suppose X is a metric space. The asymptotic dimension of X is at most n if, for every real number r > 0, there is a decomposition of X into a union of its subsets X 0 , . . . , X n such that each X i is the union of a uniformly bounded and r-disjoint family U i . That means there is a real number S > 0 with each member of U i being of diameter at most S and the distance between points belonging to different elements of U i is at least r.Since then several concepts related to asymptotic dimension were introduced by various authors. One can see them as a spectrum with asymptotic dimension being the strongest concept and weak coarse paracompactness being the weakest (see [7]). The concept closest to asymptotic dimension was introduced by Dranishnikov [9] under the name of Asymptotic Property C: Definition 1.2. Suppose X is a metric space. X has Asymptotic Property C if, for every sequence of real numbers r i > 0, there is a decomposition of X into a finite union of its subsets X 0 , . . . , X n for some natural n such that each X i is the union of a uniformly bounded and r i -disjoint family U i .

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Large scale absolute extensors

Dydak

Mitra

2016

Topology and its Applications

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This paper is devoted to dualization of dimension-theoretical results from the small scale to the large scale. So far there are two approaches for such dualization: one consisting of creating analogs of small scale concepts and the other amounting to the covering dimension of the Higson corona ν(X) of X. The first approach was used by M.Gromov when defining the asymptotic dimension asdim(X) of metric spaces X. The second approach was implicitly contained in the paper [6] by Dranishnikov on asymptotic topology. It is not known if the two approaches yield the same concept. However, Dranishnikov-Keesling-Uspenskiy proved dim(ν(X) ≤ asdim(X) and Dranishnikov established that dim(ν(X) = asdim(X) provided asdim(X) < ∞. We characterize asymptotic dimension (for spaces of finite asymptotic dimension) in terms of extensions of slowly oscillating functions to spheres. Our approach is specifically designed to relate asymptotic dimension to the covering dimension of the Higson corona ν(X) in case of proper metric spaces X. As an application, we recover the results of Dranishnikov-Keesling-Uspenskiy and Dranishnikov.

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Large Scale Versus Small Scale

Cited by 5 publications

References 42 publications

Coarse Amenability and Discreteness

Coarse Amenability and Discreteness

Matrix algebra of sets and variants of decomposition complexity

Large scale absolute extensors

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