Let the Mandelbrot set be the set of all complex parameters c∈C for which the orbit of 0 under iteration of the quadratic polynomial z↦z2+c does not escape to infinity; equivalently, it is the set of all c∈C such that for every non-negative integer k, the k-fold iterate satisfies ∥(z↦z2+c)[k](0)∥≤2, where ∥⋅∥ 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 C, is locally connected.