This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.
The Mandelbrot set is simply connected. This follows from a
theorem of Douady and Hubbard that there is a conformal isomorphism
from the complement of the Mandelbrot set to the complement of the
unit disk. (In other words, all equipotential curves are simple closed
curves.) It is conjectured that the Mandelbrot set is locally
connected, and thus pathwise connected, but this is currently
Connected: X is connected if there are no proper closed subsets A and
B of X such that A union B = X, but A intersect B is empty. I.e. X is
connected if it is a single piece.
Simply connected: X is simply connected if it is connected and every
closed curve in X can be deformed in X to some constant closed curve.
I.e. X is simply connected if it has no holes.
Locally connected: X is locally connected if for every point p in X,
for every open set U containing p, there is an open set V containing p
and contained in the connected component of p in U. I.e. X is locally
connected if every connected component of every open subset is open in
X. Arcwise (or path) connected: X is arcwise connected if every two
points in X are joined by an arc in X.
(The definitions are from "Encyclopedic Dictionary of Mathematics".)
Douady, A. and Hubbard, J., "Comptes Rendus" (Paris) 294, pp.123-126,