As always, we have to distinguish between the diagram itself, the representation or sign inscribed in some medium, and the formal object it represents under a given interpretation.
A venn diagram is an iconic sign we use to represent a formal object, namely, a universe of discourse, by virtue of properties the sign shares with the object. But it is only the relevant properties that do the job — the icon has many properties the object lacks and the object has many properties the icon lacks.
As far as the universe of discourse goes, its regions do not necessarily have any boundaries defined. In order to define boundaries for the regions we need to impose a particular topology on the object space.
However, even at the level of abstract logical properties, such as described by a propositional calculus, we can construct a differential extension of the calculus by attaching names to the qualitative changes involved in crossing from regions to their complements, and that is what leads to the simplest order of differential logic.
See the following articles for the basic intuitions.