A gravitational lens need not produce an odd number of images

Given any space-time M without singularities and any event O, there is a natural continuous mapping f of a two dimensional sphere into any space-like slice T not containing O. The set of future null geodesics (or the set of past null geodesics) forms a 2-sphere S and the map f sends a point in S to the point in T which is the intersection of the corresponding geodesic with T . To require that f , which maps a two dimensional space into a three dimensional space, satisfy the condition that any point in the image of f has an odd number of preimages, is to place a very strong condition on f . This is exactly what happens in any case where the odd image theorem holds for a transparent gravitational lens. It is argued here that this condition on f is too restrictive to occur in general; and if it appears to hold in a specific example, then some f should be calculated either analytically or numerically to provide either an illustrative example or counterexample.