  For a topological dynamical system we characterize the decomposition of the
state space induced by the fixed space of the corresponding Koopman operator.
For this purpose, we introduce a hierarchy of generalized orbits and obtain the
finest decomposition of the state space into absolutely Lyapunov stable sets.
Analogously to the measure-preserving case, this yields that the system is
topologically ergodic if and only if the fixed space of its Koopman operator is
one-dimensional.
