Mandelbrot set is connected. That is to say within a mandelbrot set for any pair of points there is a path within the set, connecting these points.
What abouthe set of all other points? Is there a pair of points that cannot be connected without going through the Mandelbrot set?
Yes, it is connected. Adriand Douady and John H. Hubbard proved that the complement of the Mandelbrot set and $\{z\in\Bbb C\mid|z|>1\}$ are conformally equivalent. In particular, they are homeomorphic. So, since $\{z\in\Bbb C\mid|z|>1\}$ is connected, the complement of the Mandelbrot set is connected too.