The Mandelbrot lemniscate domains $M_k=\\{c : |p_k(c)| \\le 2\\}$ are connected

For $c \in \mathbb{C}$ write $f_c(z) = z^2 + c$ and let

be the $k$-th point of the critical orbit, a monic-up-to-scale polynomial in $c$ of degree $2^{k-1}$ for $k \ge 1$ (with $p_0 \equiv 0$). The $k$-th Mandelbrot lemniscate domain is the filled sublevel set

The claim is that $M_k$ is connected for every $k$.

These sets are the standard finite-time approximations of the Mandelbrot set: they decrease, $M_0 \supseteq M_1 \supseteq \cdots$, and by the radius-$2$ escape criterion their intersection is exactly $M$. The first few are transparent: $M_0 = \mathbb{C}$ and $M_1$ is the closed disk of radius $2$, while for $k \ge 2$ the set $M_k$ is bounded by the lemniscate $|p_k(c)| = 2$, a curve of degree $2^{k-1}$.

Connectivity of a filled sublevel set $\{|p| \le R\}$ of a complex polynomial is governed by the critical values of $p$: the components of $\{|p| < R\}$ merge as $R$ grows past the moduli of the critical values, and the sublevel set is connected once $R$ dominates all of them. For the Mandelbrot polynomials the relevant fact is that every critical value of $p_k$ has modulus at most $2$, so no splitting occurs at level $2$. Equivalently, in the language of Douady and Hubbard, the Green's function of $M_k$ has no critical point in the region $\{|p_k| > 2\}$, which is the analytic heart of their proof that $M$ is connected.

Connectedness is stated in the preconnected form, which for these nonempty sets is equivalent to connectedness.