  This paper presents a unification and a generalization of the small-gain
theory subsuming a wide range of existing small-gain theorems. In particular,
we introduce small-gain conditions that are necessary and sufficient to ensure
input-to-state stability (ISS) with respect to closed sets. Toward this end, we
first develop a Lyapunov characterization of $\omega$ISS via finite-step
$\omega$ISS Lyapunov functions. Then, we provide the small-gain conditions to
guarantee $\omega$ISS of a network of systems. Finally, applications of our
results to partial input-to-state stability, ISS of time-varying systems,
synchronization problems, incremental stability, and distributed observers are
given.
