On Bing points in infinite-dimensional hereditarily indecomposable continua
type: Item
instance of: scholarly article
title: On Bing points in infinite-dimensional hereditarily indecomposable continua
zbMATH Open document ID: 1024.54021
DOI: 10.1016/S0166-8641(01)00216-4
author: Elżbieta Pol
author: Mirosława Reńska
published in: Topology and its Applications
publication date: 2003-03-10T00:00:00Z
review text: The main thrust of this research is to show the existence of examples of infinite-dimensional continua with certain properties. Let \(X\) be a continuum (compact, connected, metrizable space). The set \(B_\infty(X)\) of Bing points of \(X\) consists of all \(x\) having the property that any non-trivial subcontinuum \(K\) of \(X\) containing \(x\) must satisfy \(\dim K= \dim X\). The second author and \textit{R. Pol} already have studied such sets in the case that \(dim X<\infty\) [Trans. Am. Math. Soc. 354, 2921-2932 (2002; Zbl 0992.54030)]. Now the case that dim\(X=\infty\) is considered. Techniques used in this paper may be associated with D. Henderson, M. Levin, and others. But, in particular, a method due to T. Maćkoviak is employed in a strong manner. This enables one, for example, to create a space and a map of it to the pseudoarc, whose fibers consist of interesting continua all of whose points are limit points of the domain (see Theorem 3.1). Some of the results of the paper are as follows. In section 4 (Theorem 4.6) it is proved that for each countable infinite ordinal \(\alpha\), there exist hereditarily indecomposable continua \(K\), \(L\) with \(\operatorname {Ind}K= \alpha=\operatorname {ind}L\). It is not demonstrated whether this could be done with \(K=L\), but the authors' Remark 4.7 shows that the technique they employ would not imply that. Examples in section 5 concern the existence of hereditarily indecomposable continua with certain properties. Let us state two of these. Assume again that \(\alpha\) is a countable infinite ordinal. Example 5.2. For any nonempty \(0\)-dimensional compactum \(D\), there exist hereditarily indecomposable continua \(X\) and \(Y\) with \(\operatorname {Ind}X=\alpha=\operatorname {ind}Y\) such that both \(B_\infty(X)\) and \(B_\infty(Y)\) are homeomorphic to \(D\). Example 5.5. There exists a hereditarily indecomposable continuum \(Z\) such that \(B_\infty(Z)\) is a \(1\)-dimensional dense \(G_\delta\)-subset of \(Z\) …
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 1879070
zbMATH Keywords: continuum
zbMATH Keywords: indecomposable continuum
zbMATH Keywords: hereditarily indecomposable continuum
zbMATH Keywords: composant
zbMATH Keywords: countable-dimension
zbMATH Keywords: weak infinite-dimension
zbMATH Keywords: strong infinite-dimension
zbMATH Keywords: Bing point
zbMATH Keywords: property C
zbMATH Keywords: small transfinite inductive dimension
zbMATH Keywords: large transfinite inductive dimension
zbMATH Keywords: punctiform set
zbMATH Keywords: pseudosuspension
reviewed by: Leonard R. Rubin
MaRDI profile type: Publication
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full work available at URL: s0166-8641(01)00216-4
OpenAlex ID: W1985349539
description: scientific article; zbMATH DE number 1879070