I am interested in the topological dimension of the boundary $\partial M$ of the Mandelbrot set. While it is known that $\partial M$ has Hausdorff dimension 2, and $M$ has topological dimension 2 (being a compact set in $\mathbb{R}^2$ with nonempty interior), I am not so sure about $\partial M$. A theorem (Theorem 6.8.11) in Engelking, Topology, A Geometric Approach states that a set $A\subset \mathbb{R}^m$ has topological dimension $m$ iff $int (A)\neq \emptyset$, so it would be sufficient to know if the interior is empty or not.
So does anyone have an idea or a source to point to? Thank you in advance!