Openly homogeneous continua having only arcs for proper subcontinua
type: Item
instance of: scholarly article
title: Openly homogeneous continua having only arcs for proper subcontinua
zbMATH Open document ID: 0668.54021
DOI: 10.1016/0166-8641(89)90077-1
author: J. R. Prajs
published in: Topology and its Applications
publication date: 1989-01-01T00:00:00Z
review text: The author proves that if a non-degenerate metric continuum is openly homogeneous and all its non-degenerate proper subcontinua are arcs, then it is a solenoid. Recall that a space is called homogeneous if for each pair of its points there exists a homeomorphism of the space onto itself carrying one point of the pair to the other; it is called openly homogeneous if this condition is weakened to the requirement that there exists an open (continuous) mapping of the space onto itself having above property. The result of the author strengthens a result of \textit{C. L. Hagopian} [Pac. J. Math. 68, 425-435 (1977; Zbl 0364.54027)] who got the same conclusion assuming homogeneity instead of open homogeneity. The so called \(\epsilon\)-push property of homogeneous continua, a corollary of the Effros theorem, played a crucial role in obtaining this result. The author uses an anlogue of the \(\epsilon\)-push property due to \textit{J. J. Charatonik} and \textit{T. Maćkowiak} [Trans. Am. Math. Soc. 298, 579-602 (1986; Zbl 0608.54012)]. An original tool due to the author is introducing a metric \(\hat d\) associated in a special manner with the given metric d. If the continuum is openly homogeneous and all its non- degenerate proper subcontinua are arcs this metric is equivalent to the given one. However, the difference between d and \(\hat d\) is so essential that if they are equivalent and the continuum has only arcs as its non- degenerate proper subcontinua, then the continuum is a solenoid. This is the main lemma of the paper. The author considers also continua whose non-degenerate proper subcontinua are arcs if they are sufficiently small. For continua from this wider class the author proves that they are solenoids, but now the condition under which this is true is somewhat stronger, namely the homogeneity with respect to open light mappings. The problem of omitting ``light'' in the last statement remains open, although if the assumption of lightness is changed by the assu…
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54F65
zbMATH DE Number: 4094361
zbMATH Keywords: open mappings
zbMATH Keywords: light mappings
zbMATH Keywords: arc component
zbMATH Keywords: generalized homogeneity of continua
zbMATH Keywords: non-degenerate metric continuum
zbMATH Keywords: non-degenerate proper subcontinua
zbMATH Keywords: arcs
zbMATH Keywords: solenoid
zbMATH Keywords: open homogeneity
zbMATH Keywords: \(\epsilon \) -push property
reviewed by: Jerzy Mioduszewski
MaRDI profile type: Publication
full work available at URL: 0166-8641(89)90077-1
OpenAlex ID: W2051166531
cites work: A Simple Closed Curve is the Only Homogeneous Bounded Plane Continuum that Contains an Arc
cites work: Some applications of an approximation theorem for inverse limits
cites work: Confluent mappings and unicoherence of continua
cites work: Q3311300
cites work: Around Effros' Theorem
cites work: Arc components of chainable Hausdorff continua
cites work: Q4074079
cites work: A characterization of solenoids
cites work: Q3753119
cites work: Q3319407
cites work: Compositions of confluent mappings and some other classes of functions
cites work: Q3706925
cites work: Continuously Homogeneous Continua and their ARC Components
cites work: Decompositions of homogeneous continua
cites work: Q5727237
description: scientific article; zbMATH DE number 4094361