(1) The value of any sheaf on the empty set is the terminal object. (Consider the gluing condition for the empty open cover of the empty set.)
(2) If A→B is a morphism of sets, then we can define the factor set B/A. We have B/∅=B⊔*, where * is a one-element set. (Consider the left adjoint of the forgetful functor from the category of pointed sets to the category of morphisms of sets.)
(3) Sometimes the norm of a morphism of normed spaces f: X→Y is defined as sup_{x∈X: x≠0} ‖f(x)‖/‖x‖ or as sup_{x∈X: ‖x‖=1} ‖f(x)‖. This does not work for X=0. The correct definition is ‖f‖=sup_{x∈X: ‖x‖≤1} ‖f(x)‖. It also works for seminorms.
(4) The zero ring is the terminal object in the category of unital rings. It is not an integral domain, nor a local ring or a field.
(5) The empty manifold is not connected. Its number of connected components is 0. (Think of the following theorem: Every manifold is the coproduct of a unique family of connected manifolds. The cardinality of the family equals the number of connected components.)
(6) Examples in elementary mathematics abound. The zero vector space has an empty basis and a unique endomorphism A. The matrix of A in the unique basis is empty and the determinant of A is 1. There is exactly one function from the empty set to any other set (the empty function). Zero is a natural number, 0^0=1, the sum of the empty family of numbers is 0, the product of the empty family of numbers is 1, the product or the coproduct of an empty family of objects in a category is the terminal or the initial object of this category, the monoidal product of the empty family of objects in a monoidal category is the monoidal unit.
