Observability and topological dynamics (Q1175458)

A dynamical system \((X,\phi )\), continuous or discrete, \(X\) compact and metric is observable if there exists a continuous function \(f:X\to X\) such that for every two distinct points \(x, y\) one can find \(t>0\) such that \(f(\phi (x,t))\neq f(\phi (y,t))\). The author claims that if the space \(X\) either (a) is a topological manifold of finite dimension or (b) has finite covering dimension, and the set of periodic points of period \(p\) is finite for each \(p\), then \((X,\phi )\) is observable.