Complement of unit sphere is disconnected?

Question

I have proved that unit sphere is connected in $\Bbb R^3$, can I use this fact to prove that complement of unit sphere is not connected?

Answer 1

No. Instead, try assuming for contradiction that there is a path $\gamma\colon I\to\mathbb R^3$ going from, say, $\mathbf 0$ to some $x\in\mathbb R^3$ with $|x|>1$. Then use a connectedness argument to show that the continuous map $|\gamma |$ (composite of $\gamma$ and the norm $|\cdot |$) must pass through $1$. This is sufficient.

Answer 2

No, but you write the complement as the union of $\{x:||x\|<1\}$ and $\{x:||x\|>1\}$ which proves (from definition of connectedness) that it is not connected.

Note that $\{(x,y,z): x, y ,z \geq 0\}$ and its complement are both connected, so your argument does not work.