Local Connectivity of the Mandelbrot Set

Let the Mandelbrot set be the set of all complex parameters $c \in \mathbb{C}$ for which the orbit of $0$ under iteration of the quadratic polynomial $z \mapsto z^2 + c$ does not escape to infinity; equivalently, it is the set of all $c \in \mathbb{C}$ such that for every non-negative integer $k$, the $k$-fold iterate satisfies $\|(z \mapsto z^2 + c)^{[k]}(0)\| \leq 2$, where $\|\cdot\|$ denotes the complex modulus. A subset of a topological space is locally connected if every point has a neighborhood base consisting of connected open sets. Prove that the Mandelbrot set, equipped with the subspace topology from $\mathbb{C}$, is locally connected.