The problem of the formalization of constructive topology

Formal topologies are today an established topic in the development of constructive mathematics. One of the main tools in formal topology is inductive generation since it allows to introduce inductive methods in topology. The problem of inductively generating formal topologies with a cover relation and a unary positivity predicate has been solved in "Inductively generated formal topologies" by T.Coquand, G.Sambin, J.Smith and S.Valentini, Annals of Pure and Applied Logic (124). However, to deal both with open and closed subsets, a binary positivity predicate has to be considered. In this paper we will show how to adapt to this framework the method used to generate inductively formal topologies with a unary positivity predicate; the main problem that one has to face in such a new setting is that, as a consequence of the lack of a complete formalization, both the cover relation and the positivity predicate can have proper axioms.