We recall the definition of simple points which uses the digital fundamental group introduced by T.Y.Kong in [Kong89]. Then, we prove that a not less restrictive definition can be given. Indeed, we prove that there is no need of considering the fundamental group of the complement of an object in order to characterize its simple points. In order to prove this result, we do not use the fact that "the number of holes of X is equal to the number of holes in \overline{X}" which is not sufficient for our purpose but we use the linking number defined in [FoureyMalg00b]. In so doing, we formalize the proofs of several results stated without proof in the literature (Bertrand, Kong, Morgenthaler).