Another view of the coarse invariant σ

Abstract

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