Phantom maps and the Gray index

Let X be a connected CW-complex. It is shown that for suitable X and Y there is a bijection between the set Ph(X,Y) of pointed homotopy classes of phantom maps from X to Y and the extension product of inverse sequences which is another characteristic phenomenon of the previous results proved by McGibbon and Roitberg. We also study the Gray index of phantom maps. Especially, we prove a necessary and sufficient condition for the finiteness of the Gray index of phantom maps relating to the open question about phantom map theory.