= Solution
For $f_c(z)=z^2+c$, the Mandelbrot set is the set of parameters for which the critical orbit of zero is bounded. If $c$ lies outside it, the critical value $c$ is in the basin of infinity and one defines the parameter Bottcher map
$$
\Phi(c)=\phi_c(c).
$$
The dynamical functional equation, holomorphic dependence on $c$, and the normalization at infinity show that $\Phi$ is a proper degree-one holomorphic map
$$
\mathbb C\setminus M\longrightarrow\mathbb C\setminus\overline{\mathbb D}.
$$
It is therefore a conformal isomorphism. Since the exterior disc is connected and simply connected, the complement of $M$ has no bounded component; equivalently, the <Mandelbrot set connectedness from the parameter Böttcher coordinate> proves that $M$ is connected.
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