  In this paper, we deal with the part of Fractal Theory related to finite
families of (weak) contractions, called iterated function systems (IFS,
herein). An attractor is a compact set which remains invariant for such a
family. Thus, we consider spaces homeomorphic to attractors of either IFS or
weak IFS, as well, which we will refer to as Banach and topological fractals,
respectively. We present a collection of counterexamples in order to show that
all the presented definitions are essential, though they are not equivalent in
general.
