Structuralist praxeologies in the perspective of Klein: the case of connectedness in analysis

This article is a continuation of our previous work on structuralist aspects of analysis at university in the framework of the Anthropological Theory of the Didactic. We now examine how the teaching of abstract structures (in particular metric spaces and topology) may contribute to Klein's perspective of “Elementary Mathematics from a higher standpoint”. Precisely, on the basis of a real and abstract analysis textbook used in the transition between Bachelor and Master degree programs in mathematics and in the light of the notion of structuralist praxeology and its dialectics, we discuss whether we can defend teaching the notion of connectedness to teacher students as a means to link real and abstract analyses in the spirit of Klein.