  This paper presents reduction theorems for stability, attractivity, and
asymptotic stability of compact subsets of the state space of a hybrid
dynamical system. Given two closed sets $\Gamma_1 \subset \Gamma_2 \subset
\Re^n$, with $\Gamma_1$ compact, the theorems presented in this paper give
conditions under which a qualitative property of $\Gamma_1$ that holds relative
to $\Gamma_2$ (stability, attractivity, or asymptotic stability) can be
guaranteed to also hold relative to the state space of the hybrid system. As a
consequence of these results, sufficient conditions are presented for the
stability of compact sets in cascade-connected hybrid systems. We also present
a result for hybrid systems with outputs that converge to zero along solutions.
If such a system enjoys a detectability property with respect to a set
$\Gamma_1$, then $\Gamma_1$ is globally attractive. The theory of this paper is
used to develop a hybrid estimator for the period of oscillation of a
sinusoidal signal.
