Singular homology of roots of unity

We prove analogues of fundamental results from algebraic topology in the setting of \v{C}ech's closure spaces. For a singular homology theory of closure spaces we prove analogues of the excision and Mayer-Vietoris theorems. Furthermore, we prove a Hurewicz theorem in dimension one. We use these results to calculate examples of singular homology groups of spaces that are not topological but one often encounters in applied topology, such as simple undirected graphs. We mainly focus on singular homology of roots of unity with closure structures arising from considering nearest neighbors. These ''simpler'' closure spaces can then serve as building blocks along with our Mayer-Vietoris and excision theorems for calculating the singular homology of more complex closure spaces.