The Mandelbrot set is a set of complex numbers \( c \) for which the function \( f_c(z) = z^2 + c \) does not diverge when iterated from \( z = 0 \): \[ M = \{c \in \mathbb{C} : \forall n \in \mathbb{N}, |f_c^n(0)| \leq 2 \} \]