Let T be a self homeomorphism of a compact metric space X. Suppose that \({\mathcal E}=\{E_ 1,E_ 2,...\}\) is a sequence of Borel subsets of X. \({\mathcal E}\) is said to be a separator if \(\lim_{n\to \infty}E_ n\) has invariant measure zero and if \({\mathcal E}'\) is an infinite sub-collection of sets from \({\mathcal E}\) and x, y are distinct points of X, then there is an integer k and a set \(E\in {\mathcal E}'\) which separates \(T^ kx\) from \(T^ ky\). The major theorem of the paper asserts that if (X,T) admits a separator \({\mathcal E}\), then the topological entropy of T is zero.