In a G-space R if B is a co-ray
to A then the union of all co-rays to A that contain B is either a straight line or a
co-ray to A maximal in that it is properly contained in no other co-ray to A. In the
latter case, the initial point of the maximal co-ray is a copoint to A. The concept of
co-point is an analogue to that of minimum point in a sense made precise. On certain
non-compact G-surfaces of finite connectivity, including those with non-positive
curvature, we characterize the locus of co-points to a given ray and obtain bounds for
the number of components of this locus, the number of co-rays emanating from a
co-point and the number of co-points that are origins of more than two
co-rays.
