Show that $M \times \left\{0,1 \right\}$ contains two connected components

Question

Let $M$ be a smooth manifold (without boundary). Prove that $M \times \left\{0,1 \right\}$ contains two connected components, each of which is diffeomorphic to M.

I've addressed the problem when $M$ is connected. Could anyone give me ideas in the case that $M$ is disconnected? Any help would be appreciated.

Answer 1

$M\times\{0,1\}$ is diffeomorphic to $M\sqcup M$, so the number of path connected components of $M\times\{0,1\}$ is exactly twice as that of $M$.