Characterizing Lipschitz images of injective metric spaces. arXiv:2405.01860
Preprint, arXiv:2405.01860 [math.GN] (2024).
Summary: A metric space \(X\) is {\em injective} if every non-expanding map \(f:B\to X\) defined on a subspace \(B\) of a metric space \(A\) can be extended to a non-expanding map \(\bar f:A\to X\). We prove that a metric space \(X\) is a Lipschitz image of an injective metric space if and only if \(X\) is Lipschitz connected in the sense that for every points \(x,y\in X\), there exists a Lipschitz map \(f:[0,1]\to X\) such that \(f(0)=x\) and \(f(1)=y\). In this case the metric space \(X\) carries a well-defined intrinsic metric. A metric space \(X\) is a Lipschitz image of a compact injective metric space if and only if \(X\) is compact, Lipschitz connected and its intrinsic metric is totally bounded. A metric space \(X\) is a Lipschitz image of a separable injective metric space if and only if \(X\) is a Lipschitz image of the Urysohn universal metric space if and only if \(X\) is analytic, Lipschitz connected and its intrinsic metric is separable.
MSC:
54E35 | Metric spaces, metrizability |
54E40 | Special maps on metric spaces |
51F30 | Lipschitz and coarse geometry of metric spaces |
54C55 | Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties) |
54E45 | Compact (locally compact) metric spaces |
54E50 | Complete metric spaces |
54F15 | Continua and generalizations |
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