Characterizing Lipschitz images of injective metric spaces
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909210877362176 |
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| author | Bąk, Judyta Banakh, Taras Garbulińska-Węgrzyn, Joanna Nowak, Magdalena Popławski, Michał |
| author_facet | Bąk, Judyta Banakh, Taras Garbulińska-Węgrzyn, Joanna Nowak, Magdalena Popławski, Michał |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01860 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterizing Lipschitz images of injective metric spaces Bąk, Judyta Banakh, Taras Garbulińska-Węgrzyn, Joanna Nowak, Magdalena Popławski, Michał General Topology Primary:54E35, 54E40, Secondary: 51F30, 54C55, 54E45, 54E50, 54F15 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. |
| title | Characterizing Lipschitz images of injective metric spaces |
| topic | General Topology Primary:54E35, 54E40, Secondary: 51F30, 54C55, 54E45, 54E50, 54F15 |
| url | https://arxiv.org/abs/2405.01860 |