  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.
