On the formal points of the formal topology of the binary tree

Formal topology is today an established topic in the development of constructive mathematics and constructive proofs for many classical results of general topology have been obtained by using this approach. Here we analyze one of the main concepts in formal topology, namely, the notion of formal point. We will contrast two classically equivalent definitions of formal points and we will see that from a constructive point of view they are completely different. Indeed, according to the first definition the formal points of the formal topology of the real numbers can be indexed by a set whereas this is not possible according to the second one.