I've wondered, is the Mandelbrot set one continuous shape, or is broken into many smaller shapes? Put crudely, if it was imported into a paint program, and you performed a fill action, would the whole shape change to a single color?
> Douady and Hubbard have shown that the Mandelbrot set is connected. In fact, they constructed an explicit conformal isomorphism between the complement of the Mandelbrot set and the complement of the closed unit disk.
Not sure why the existence of this isomorphism implies it is connected, but there you go
