Lyapunov functions and attractors in arbitrary metric spaces

We prove two theorems concerning Lyapunov functions on metric spaces. The new element in these theorems is the lack of a hypothesis of compactness or local compactness. The first theorem applies to a discrete dynamical system on any metric space; the result is that if $A$ is an attractor for a continuous map $g$ of a metric space $X$ to itself, then there is a Lyapunov function for $A$. The second theorem applies only to separable metric spaces; the theorem is that there is a complete Lyapunov function for any continuously-generated discrete dynamical system on a separable metric space. (A complete Lyapunov function is a real-valued function that is constant on orbits in the chain recurrent set, is strictly decreasing along all other orbits, and separates different components of the chain recurrent set.)