**Motivation**: The Mandlebrot set is a simply connected set with an infinitely complex boundary, but CAN one move from interior to the exterior of this topological space by just crossing over a finite set of points?

**Application**: I am modeling the leakage of emission of EM waves, through a fractal forest where dispersion occurs every time an edge of the forest is crossed

Let $\mathbb{M}\subset \mathbb{C}$ be the Mandelbrot Set. Let $\partial \mathbb{M}\subset \mathbb{M}$ be its boundary. Consider two points $\mathbb{z_1}\in \mathbb {M}$ and $\mathbb{z_2}\in\mathbb{C}\notin \mathbb{M}$

My **question** is:

does there exist a path $\mathbb{P}$ $ \subset \mathbb{C}$ with initial point $\mathbb{z_1}$ and terminal point $\mathbb{z_2}$?

Is $\mathbb{P}\bigcap\partial\mathbb{M}$ a finite set?