Iterated quasicomponents of subspaces of rational continua
type: Item
instance of: scholarly article
title: Iterated quasicomponents of subspaces of rational continua
zbMATH Open document ID: 0783.54019
DOI: 10.1016/0166-8641(93)90141-Y
author: Eric K. van Douwen
published in: Topology and its Applications
publication date: 1993-11-15T00:00:00Z
review text: A space \(X\) is called rational if it has a base of open sets each having a boundary which is countable. For any \(x\in X\) define the index of nonconnectedness of \(X\) at \(x\), \(nc(X,x)\), to be the smallest ordinal \(\alpha\) such that the component of \(x\) at \(X\) is obtained by iterating the taking of the quasicomponents at \(x\), \(\alpha\) many times. \textit{A. Lelek} [Fundam. Math. 67, 359-367 (1970; Zbl 0201.248)] has proved that if \(X\) is a rational metrizable continuum, then \(\sup\{nc(Y,x): x\in Y\}<\omega_ 1\) for all \(Y\subset X\) and has asked if this can be improved to \(\sup\{nc(Y,x): x\in Y\subset X\}<\omega_ 1\). In this article, the author has constructed a rational planar continuum \(X\) which is the countable union of straight-line segments having a common end- point and with a point \(x\) for which \(\sup\{nc(Y,x): x\in Y\subset X\}= \omega_ 1\).
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 446208
zbMATH Keywords: rational space
zbMATH Keywords: index of nonconnectedness
zbMATH Keywords: component
zbMATH Keywords: quasicomponents
zbMATH Keywords: rational planar continuum
reviewed by: Richard G. Wilson
MaRDI profile type: Publication
cites work: On the topology of curves I
cites work: Some problems concerning curves
full work available at URL: 0166-8641(93)90141-y
OpenAlex ID: W2091534971
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description: scientific article; zbMATH DE number 446208