Consider a 2-dimensional ODE in the real plane. An equivalence relation is defined among the points a, b in the plane :

aRb if an only if there is a trajectory connecting a and b.

The space of all the equivalence class might be thought as a subset of the real line : is it possible to build this set and give it a separable topology or a metric ? One problem is to choose a representative for each class so as to get to define the distance between two solutions of an ODE...