Hadwiger's theorem characterizes this quantity as the measure for the finite union of compact convex sets that is homogeneous degree zero. For a chain complex, it is defined as the alternating sum of the ranks of the homology groups, and can be defined as the alternating sum of Betti numbers. The this quantity is proportional to the integral of the curvature of a closed Riemannian manifold according to the Gauss-Bonnet theorem. The only closed compact surfaces for which 
Hadwiger's theorem characterizes this quantity as the measure for the finite union of compact convex sets that is homogeneous degree zero. For a chain complex, it is defined as the alternating sum of the ranks of the homology groups, and can be defined as the alternating sum of Betti numbers. The this quantity is proportional to the integral of the curvature of a closed Riemannian manifold according to the Gauss-Bonnet theorem. The only closed compact surfaces for which it is zero are the torus and the Klein bottle, and its namesake's formula states that it is equal to two for the surface of any convex polyhedron. For 10 points, name this topological invariant, denoted by the letter chi, which was defined by its Swiss namesake as vertices minus edges plus faces.
Hadwiger's theorem characterizes this quantity as the measure for the finite union of compact convex sets that is homogeneous degree zero.
Euler characteristic [or Euler number]