This paper aims at defining and exploring design principles in a distance technological setting for an educational activity for mathematics undergraduate students, devoted to the construction of basic concepts in general topology, the promotion of problem-solving processes, the development of metacognitive aspects, and, in general, the development of the students' mathematical identity. The design exploits the production of examples and investigation of variations and invariants, exploration of problems and generation of conjectures, and an extension intertwining of the 'inside-out' model from the Digital Interactive Storytelling in Mathematics with the Thinking Classroom model at university education. We present a didactic activity based on the identified design principles and discuss the preliminary results of a pilot carried out with fifty mathematics undergraduate students, attending their second year of the mathematics degree.