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Some results on maps that factor through a tree. (English) Zbl 1317.51008
A tree is a metric space \(T\) that is uniquely arc-connected, i.e., for different points \(p, p' \in T \) there is an embedding \(\gamma: [0,1]\rightarrow T\) with \(\gamma(0)=p, \gamma(1)=p'\) and any other such embedding is a reparameterization of \(\gamma\).
Let \(\varphi : X \rightarrow Y\) be a uniformly continuous map between metric spaces. The mapping \(\varphi\) is said to have Property (T) if for all \(x, x' \in X\) with \(\varphi (x) \neq \varphi(x')\) there is a point \(y \in Y \backslash \{ \varphi(x), \varphi(x')\}\) such that for any curve \(\gamma: [0, 1] \rightarrow X\) connecting \(x\) with \(x', y\) is contained in \(\text{im}(\varphi\circ\gamma)\).
The author characterizes maps \(\varphi\) which factor through a tree. The Property (T) is frequently used throughout the paper. Theorem 1.2 gives a characterization of Property (T).

51F99 Metric geometry
30L99 Analysis on metric spaces
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
55M25 Degree, winding number
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