A nonstandard invariant of coarse spaces

Abstract

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 $\le 1$ is provided. Miller et al. 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 $\left(X,\xi \right)$.

References in corpus (3)

• Nonstandard methods in large-scale topology
• Sequential ends of metric spaces
• Nonstandard homology theory for uniform spaces