  Euclid defined a boundary as “that which is an extremity of anything” (Elements Bk I, Df 13), and Aristotle made this more precise by defining the extremity of a thing x as “the first point beyond which it is not possible to find any part [of x], and the first point within which every part [of x] is.” (Metaphysics V, 17, 1022a4–5) This definition is intuitive enough and may be regarded as the natural starting point for any investigation into the concept of a boundary. Indeed, although Aristotle's definition was only meant to apply to material substances, it intuitively applies to events as well (insofar as they have mereological structure) and by extension also to abstract entities such as concepts and sets (compare the topologically standard notion of the boundary of a set x as the set of those points all of whose neighborhoods intersect both x and the complement of x). Nonetheless, precisely this intuitive characterization gives rise to several puzzles that justify philosophical concern, especially with respect to the boundaries of spatio-temporal particulars such as objects and events.