Another view of the coarse invariant $\sigma$

Abstract

Miller, Stibich and Moore (2010) developed a set-valued coarse invariant $\sigma \left(X,\xi \right)$ of pointed metric spaces. DeLyser, LaBuz and Tobash (2013) provided a different way to construct $\sigma \left(X,\xi \right)$ (as the set of all sequential ends). This paper provides yet another definition of $\sigma \left(X,\xi \right)$. To do this, we introduce a metric on the set $S\left(X,\xi \right)$ of coarse maps $\left(\mathbb{N},0\right)\to \left(X,\xi \right)$, and prove that $\sigma \left(X,\xi \right)$ is equal to the set of coarsely connected components of $S\left(X,\xi \right)$. As a by-product, our reformulation trivialises some known theorems on $\sigma \left(X,\xi \right)$, including the functoriality and the coarse invariance.