
doi: 10.4064/cm97-1-6
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.
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
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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