Abstract. It is well known that for a transitive dynamical system (X, f ) sensitivity to initial conditions follows from the assumption that the periodic points are dense. This was done by several authors: Banks, Brooks, Cairns, Davis and Stacey [2], Silverman [8] and Glasner and Weiss [6]. In the latter article Glasner and Weiss established a stronger result (for compact metric systems) which implies that a transitive non-minimal compact metric system (X, f ) with dense set of almost periodic points is sensitive. This is true also for group actions as was proved in the book of Glasner [4].Our aim is to generalize these results in the frame of a unified approach for a wide class of topological semigroup actions including one-parameter semigroup actions on Polish spaces.