Do we have a good mathematical definition of connectedness?

Pascal Préa (LIS)

Summary

Mathematics can be used, amongst other things, to model the world. For example, connectedness (for which the intuitive definition is “made up of only one part”) is described by “not being partitioned into two open sets”. On the real line, the usual topology allows to recover the “usual” notion of connectedness.

We will show in this talk that this is not the case on graphs (even though for these, a rigorous definition of connectedness exists): in general there does not exist a topology on the set of vertices such that the connected components are exactly those which are not partitioned into two open sets. At the end of the talk, we will propose an alternative (mathematical) definition of connectedness, as well as linking it to the classical definition.