James mentions in the intro to his Poincaré Project that
In mathematics in general, and in the early days of this Poincare Project in particular, we will often be asking questions like: what is the most general object that exhibits this characteristic? What is the distinguishing characteristic of this object compared with others we’re dealing with?
Going meta: when may we define the open balls by saying "when two objects a and b share a most general object c, they are in the same open ball"?