Home > Journals > TAG > Vol. 5 > Iss. 2 (2018)

## Abstract

Let *(X, d)* be an unbounded metric space and let *\tilde r=(r_n)_{n\in\mathbb N}* be a sequence of positive real numbers tending to infinity. A pretangent space *\Omega_{\infty, \tilde r}^{X}* to *(X, d)* at infinity is a limit of the rescaling sequence *\left(X, \frac{1}{r_n}d\right)*. The set of all pretangent spaces *\Omega_{\infty, \tilde r}^{X}* is called an asymptotic cluster of pretangent spaces. Such a cluster can be considered as a weighted graph *(G_{X, \tilde r}, \rho_{X})* whose maximal cliques coincide with *\Omega_{\infty, \tilde r}^{X}* and the weight *\rho_{X}* is defined by metrics on *\Omega_{\infty, \tilde r}^{X}*. We describe the structure of metric spaces having finite asymptotic clusters of pretangent spaces and characterize the finite weighted graphs which are isomorphic to these clusters.

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

## Recommended Citation

Bilet, Viktoriia and Dovgoshey, Oleksiy
(2018)
"Finite Asymptotic Clusters of Metric Spaces,"
*Theory and Applications of Graphs*: Vol. 5:
Iss.
2, Article 1.

DOI: 10.20429/tag.2018.050201

Available at:
https://digitalcommons.georgiasouthern.edu/tag/vol5/iss2/1

*Supplemental file with DOI*