The index of sources and sinks is +1, and that of saddle points is −1. So the Euler characteristic equals the difference between the number of positive indices n + and the number of negative indices n − − . For the far field case, there is only one image, i.e., χ χ = n + − − n − − = 1 . So the total number of images is N = n + + n − − = 2 n − − + 1 , i.e., odd. The strict proof needs Uhlenbeck's Morse theory of null geodesics.