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Colloquium Mathematicum
Article . 2003 . Peer-reviewed
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Arc property of Kelley and absolute retracts for hereditarily unicoherent continua

Authors: Charatonik, Janusz J.; Charatonik, Włodzimierz J.; Prajs, Janusz R.;

Arc property of Kelley and absolute retracts for hereditarily unicoherent continua

Abstract

If \({\mathcal K}\) is a class of compact metric spaces, then \(\text{AR}({\mathcal K})\) denotes the family of all absolute retracts for \({\mathcal K}\), i.e. \(K\in\text{AR}({\mathcal K})\) provided that if \(Z\in{\mathcal K}\) contains a homeomorphic copy \(K'\) of \(K\), then \(K'\) is a retract of \(Z\). A metric continuum \(X\) is said to have the arc property of Kelley provided that for each point \(p\in X\), for each subcontinuum \(K\) of \(X\) containing \(p\) and for each sequence of points \(p_n\in X\) converging to \(p\) there exists a sequence of arcwise connected subcontinua \(K_n\) of \(X\) containing \(p_n\) and converging to the continuum \(K\). In this paper the authors prove that absolute retracts for some classes of metric continua (including hereditarily unicoherent continua, tree-like continua, \(\lambda\)-dendroids and dendroids) share some basic properties with regular absolute retracts for all compacta. In particular, they show that each member of the mentioned classes has the arc property of Kelley.

Keywords

Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties), Continua and generalizations, tree-like, Topological spaces of dimension \(\leq 1\); curves, dendrites, decomposable, arc approximation property, dendroid, unionable

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
8
Average
Top 10%
Average
bronze