  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, Silverman, and also by Glasner and Weiss. 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 a recent book of Glasner. 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.
