  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.
