Given an unforced nonlinear system and two nested closed and invariant sets ¿ ¿ O, we present reduction principles allowing one to extrapolate the properties of stability, attractivity, and asymptotic stability of ¿ from analogous properties of the system restricted to O. As a corollary to our reduction principles, we present a stability criterion for cascade-connected systems which generalizes well-known results in the literature. Using the reduction principles, in Part II of this paper we present a comprehensive theory for passivity-based stabilization of closed sets.