Weak attractors from Lyapunov functions

References (Hurley, 1991, 1992, 1998) show that if a continuous map f on a metric space X has a “weak attractor”, A, then there is an associated Lyapunov function, h, which is a continuous, nonnegative, real-valued map whose zero set is A, and satisfying h∘f−h<0 on a certain deleted neighborhood of A. In (1996) Kim et al. show that If X is locally compact and if the zero set Z of a Lyapunov function is compact, then Z is a weak attractor. Here we obtain the same result without the compactness assumption on Z, provided that the ambient space is σ-compact.