We recall a possible definition of a simple point which uses the digital fundamental group introduced by T. Y. Kong in [Kong89]. Then, we prove that a more concise but not less restrictive definition can be given. Indeed, we prove that there is no need to consider 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 tunnels of $X$ is equal to the number of tunnels in $\overline{X}$'' 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).