  In this note a notion of generalized topological entropy for arbitrary
subsets of the space of all sequences in a compact topological space is
introduced. It is shown that for a continuous map on a compact space the
generalized topological entropy of the set of all orbits of the map coincides
with the classical topological entropy of the map. Some basic properties of
this new notion of entropy are considered; among them are: the behavior of the
entropy with respect to disjoint union, cartesian product, component
restriction and dilation, shift mapping, and some continuity properties with
respect to Vietoris topology. As an example, it is shown that any self-similar
structure of a fractal given by a finite family of contractions gives rise to a
notion of intrinsic topological entropy for subsets of the fractal. A
generalized notion of Bowen's entropy associated to any increasing sequence of
compatible semimetrics on a topological space is introduced and some of its
basic properties are considered. As a special case for $1\leq p\leq\infty$ the
Bowen $p$-entropy of sets of sequences of any metric space is introduced. It is
shown that the notions of generalized topological entropy and Bowen
$\infty$-entropy for compact metric spaces coincide.
