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 we deduce for example that infinitely renormalizable 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 Preptint 1998/13b.