Semi-coarse Spaces, Homotopy and Homology

We begin the study the algebraic topology of semi-coarse spaces, which are generalizations of coarse spaces that enable one to endow non-trivial `coarse-like’ structures to compact metric spaces, something which is impossible in coarse geometry. We first study homotopy in this context, and we then construct homology groups which are invariant under semi-coarse homotopy equivalence. We further show that any undirected graph $G=(V,E)$ induces a semi-coarse structure on its set of vertices $V_G$, and that the respective semi-coarse homology is isomorphic to the Vietoris-Rips homology. This, in turn, leads to a homotopy invariance theorem for the Vietoris-Rips homology of undirected graphs.