Connectedness properties of the Higson corona of the half line
type: Item
instance of: scholarly article
title: Connectedness properties of the Higson corona of the half line
zbMATH Open document ID: 1477.54026
author: Takamitsu Yamauchi
published in: Topology and its Applications
publication date: 2021-09-27T00:00:00Z
review text: The Higson corona of an unbounded proper metric space \(X\) is the remainder \(vX=hX\setminus X\) of the Higson compactification \(hX\) of \(X\). A definition of the Higson compactification can be found, for instance, in [\textit{J. Roe}, Coarse cohomology and index theory on complete Riemannian manifolds. Providence, RI: American Mathematical Society (AMS) (1993; Zbl 0780.58043)] and [\textit{J. Roe}, Lectures on coarse geometry. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1042.53027)]. In this article, the author considers in ZFC the half-line \(\mathbb{H}=\{x\in\mathbb{R}: x\geq 0\}\) equipped with the usual metric \(d\) defined by \(d(x,y)=|x-y|\) for \(x,y\in\mathbb{H}\). Using methods shown in [\textit{K. P. Hart}, in: Recent progress in general topology. Papers from the Prague Toposym 1991, held in Prague, Czechoslovakia, Aug. 19-23, 1991. Amsterdam: North-Holland. 317--352 (1992; Zbl 0847.54028)], the author obtains several interesting results concerning connectedness properties of the Higson corona \(v\mathbb{H}\) in ZFC. In particular, it is proved that \(v\mathbb{H}\) is hereditarily unicoherent, and no non-degenerate subcontinuum of \(v\mathbb{H}\) is hereditarily indecomposable. Furthermore, it is proved that the Continuum Hypothesis implies that the set of all composants of \(v\mathbb{H}\) is of cardinality \(2^{\mathfrak{c}}\). Similarly to the known fact that the Principle of Near Coherence of Filters (denoted by NCF) is equivalent to the statement that \(\beta\mathbb{H}\setminus\mathbb{H}\) has only one composant (see [\textit{A. Blass}, Trans. Am. Math. Soc. 300, 557--581 (1987; Zbl 0647.03043)] and compare with [\textit{J. Mioduszewski}, Proc Conf., Zinnowitz 1974, 257--283 (1978; Zbl 0407.54018)]), it is shown that NCF is equivalent to the assertion that \(v\mathbb{H}\) has exactly one composant. Standard subcontinua of \(v\mathbb{H}\) and layers in them are defined, and it is proved that every layer in a standard subcontinuum …
reviewed by: Eliza Wajch
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54D40
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 51F99
zbMATH DE Number: 7401527
zbMATH Keywords: Higson corona
zbMATH Keywords: hereditarily unicoherent continuum
zbMATH Keywords: indecomposable continuum
zbMATH Keywords: composant
MaRDI profile type: Publication
full work available at URL: j.topol.2020.107542
OpenAlex ID: W3110777610
cites work: On character of points in the Higson corona of a metric space.
cites work: Q3425064
cites work: A non-metric indecomposable continuum
cites work: Near coherence of filters. I: Cofinal equivalence of models of arithmetic
cites work: Near Coherence of Filters. II: Applications to Operator Ideals, the Stone-Cech Remainder of a Half-Line, Order Ideals of Sequences, and Slenderness of Groups
cites work: Asymptotic topology
cites work: On the Higson corona of uniformly contractible spaces
cites work: Q3994935
cites work: Rings of Continuous Functions in Which Every Finitely Generated Ideal is Principal
cites work: Q4281185
cites work: Higson compactifications obtained by expanding and contracting the half-open interval
cites work: Q4870666
cites work: Subcontinua of the Higson corona
cites work: Q3304890
cites work: A topological property enjoyed by near points but not by large points
cites work: Q4194097
cites work: Q3906757
cites work: Metric spaces, convexity and nonpositive curvature
cites work: Q3094509
cites work: Coarse cohomology and index theory on complete Riemannian manifolds
cites work: Q4437801
cites work: Composants and βN
cites work: β([ 0, ∞)) Does not Contain Nondegenerate Hereditarily Indecomposable Continua
cites work: Layers of Components of β([ 0, 1] × N) are Indecomposable
cites work: On indecomposable subcontinua of \(\beta{}[0,\infty{})-[0,\infty{})\)
cites work: Continua in \(R^*\)
DOI: 10.1016/J.TOPOL.2020.107542
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description: scientific article; zbMATH DE number 7401527