# A nonstandard invariant of coarse spaces

Takuma Imamura – GJM, Volume 5, Issue 1 (2020), 1-8.

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- octobre 28, 2020
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We construct a set-valued invariant \iota\left(X,\xi\right) of pointed coarse spaces \left(X,\xi\right) by using nonstandard analysis. The invariance under coarse equivalence is established. A sufficient condition for the invariant to be of cardinality \leq1 is provided. Miller *et al*. [15] and subsequent researchers have introduced a similar but standard set-valued coarse invariant \sigma\left(X,\xi\right) of pointed metric spaces \left(X,\xi\right). In order to compare these two invariants, we construct a natural transformation \omega_{\left(X,\xi\right)} from \sigma\left(X,\xi\right) to \iota\left(X,\xi\right). The surjectivity of \omega_{\left(X,\xi\right)} is proved for all proper geodesic spaces \left(X,\xi\right).

#### Milestones:

Received: September 26, 2019.

Accepted: February 22, 2020.

Published online: April 3, 2020.

#### Authors:

__Takuma Imamura__

RIMS, Kyoto University, Kitashirakawa Oiwake-cho,

Sakyo-ku, Kyoto 606-8502, Japan.