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arXiv · 2106.00049

On equivalence of unbounded metric spaces at infinity

Viktoriia Bilet·Oleksiy Dovgoshey
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Abstract

Let $(X, d)$ be an unbounded metric space. To investigate the asymptotic behavior of $(X, d)$ at infinity, one can consider a sequence of rescaling metric spaces $(X, \frac{1}{r_n} d)$ generated by given sequence $(r_n)_{n \in \mathbb N}$ of positive reals with $r_n \to \infty$. Metric spaces that are limit points of the sequence $(X, \frac{1}{r_n} d)_{n \in \mathbb N}$ will be called pretangent spaces to $(X, d)$ at infinity. We found the necessary and sufficient conditions under which two given unbounded subspaces of $(X, d)$ have the same pretangent spaces at infinity. In the case when $(X, d)$ is the real line with Euclidean metric, we also describe all unbounded subspaces of $(X, d)$ isometric to their pretangent spaces.

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Viktoriia Bilet, Oleksiy Dovgoshey. 2021-06-20. On equivalence of unbounded metric spaces at infinity. https://arxiv.org/abs/2106.00049

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