Since "C"−1 = 0, every 0-chain is a cycle (i.e. "Z"0 = "C"0); moreover, the group "B"0 of the 0-boundaries is generated by the three elements on the right of these equations, creating a two-dimensional subgroup of "C"0. So the 0th homology group "H"0("S") = "Z"0/"B"0 is isomorphic to Z, with a basis given (for example) by the image of the 0-cycle ("v"0). Indeed, all three vertices become equal in the quotient group; this expresses the fact that "S" is connected.