Any disjoint union of two or more nonempty graphs is necessarily disconnected. Every nonempty, paracompact, connected 1-manifold is homeomorphic either to R or the circle. It follows that every nonempty periodic semigroup has at least one idempotent. A subset is dense if and only if every nonempty open subset intersects it. Let M be a compact, irreducible 3-manifold with nonempty boundary. Each nonempty, closed, and bounded subset formula 40 admits a minimal element formula 41. The subdifferential is always a nonempty convex compact set. If it is finite and nonempty, then it must contain at least one idempotent. The Cartesian product of an infinite set and a nonempty set is infinite. A class is stationary if it has a nonempty intersection with every closed unbounded class. The intersection of finite but nonempty family of absorbing sets is absorbing. The only nonempty convex open set in is the entire space . Thus, , which contradicts "X" being a nonempty subset of N. We wish to show that the cartesian product of these sets is nonempty. Directed sets are a generalization of nonempty totally ordered sets.