We give new proofs that the Mandelbrot set is locally connected at every Misiurewicz point and at every point on the boundary of a hyperbolic component. The idea is to show "shrinking of puzzle pieces" without using specific puzzles. Instead, we introduce fibers of the Mandelbrot set and show that fibers of certain points are "trivial", i.e., they consist of single points. This implies local connectivity at these points.

Locally, triviality of fibers is strictly stronger than local connectivity. Local connectivity proofs in holomorphic dynamics often actually yield that fibers are trivial, and this extra knowledge is sometimes useful. We include the proof that local connectivity of the Mandelbrot set implies density of hyperbolicity in the space of quadratic polynomials.

We write our proofs more generally for Multibrot sets, which are the loci of connected Julia sets for polynomials of the form $z\mapsto z^d+c$.

Although this paper is a continuation of preprint 1998/12, it has been written so as to be independent of the discussion of fibers of general compact connected and full sets in $\mathbb{C}$ given there.