  A frequent problem in holomorphic dynamics is to prove local connectivity of
Julia sets and of many points of the Mandelbrot set; local connectivity has
many interesting implications. The intention of this paper is to present a new
point of view for this problem: we introduce fibers of these sets, and the goal
becomes to show that fibers are ``trivial'', i.e. they consist of single
points. The idea is to show ``shrinking of puzzle pieces'' without using
specific puzzles. This implies local connectivity at these points, but
triviality of fibers is a somewhat stronger property than local connectivity.
Local connectivity proofs in holomorphic dynamics often actually yield that
fibers are trivial, and this extra knowledge is sometimes useful.
  Since we believe that fibers may be useful in further situations, we discuss
their properties for arbitrary compact connected and full sets in the complex
plane. This allows to use them for connected filled-in Julia sets of
polynomials, and polynomials of the form $z^d+c$ have the property that the
impression of any dynamic ray at a rational angle is a single point. An
appendix reviews known topological properties of compact, connected and full
sets in the plane.
  The definition of fibers grew out of a new brief proof that the Mandelbrot
set is locally connected at every Misiurewicz point and at every point on the
boundary of a hyperbolic component. This proof works also for ``Multibrot
sets'', which are the higher degree cousins of the Mandelbrot set. These sets
are discussed in a self-contained sequel (IMS Preprint 1998/13a). Finally, we
relate triviality of fibers to tuning and renormalization in IMS Preprint
1998/13b.
