Is the Mandelbrot set connected?
A6h: 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 unproved.Connectedness definitions: 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
