Abstract
Let X be an unbounded metric space, and let \( \tilde{r} \) be a sequence of positive real numbers tending to infinity. We define the pretangent space \( {\Omega}_{\infty, \tilde{r}}^X \) to X at infinity as a metric space whose points are the equivalence classes of sequences \( \tilde{x}\subset X \) which tend to infinity with the rate \( \tilde{r} \). It is proved that all pretangent spaces are complete and, for every finite metric space Y, there is an unbounded metric space X such that Y and \( {\Omega}_{\infty, \tilde{r}}^X \) are isometric for some \( \tilde{r} \). The finiteness conditions of \( {\Omega}_{\infty, \tilde{r}}^X \) are completely described.
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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 15, No. 4, pp. 448–474 October–December, 2018.
The publication contains the results of studies conducted by a grant of the President of Ukraine for competitive projects (project F75/28173) of the State Fund for Fundamental Research. The first author was also supported by the National Academy of Sciences of Ukraine in the frame of the scientific research project for young scientists “Geometric properties of metric spaces and mappings in Finsler manifolds”.
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Bilet, V., Dovgoshey, O. Finite Spaces Pretangent to Metric Spaces at Infinity. J Math Sci 242, 360–380 (2019). https://doi.org/10.1007/s10958-019-04483-8
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DOI: https://doi.org/10.1007/s10958-019-04483-8