For a closed surface S, this is a topological invariant. It is given by the formula χ‎(S) = V − E + F, where V, E, and F are the numbers of vertices, edges, and faces, respectively, in a subdivision of S. For any surface homeomorphic to a sphere χ‎(S) = 2 (see Euler’s theorem). Poincaré generalized this to other closed surfaces: χ‎ = 2−2g for a torus with g holes and χ‎ = 2−k for a sphere with k > 0 Möbius strips sewn in. See Classification Theorem for Surfaces, Gauss-Bonnet theorem, poincaré-hopf theorem.