Complement of unit sphere is disconnected?

by GiannaG   Last Updated May 16, 2019 04:20 AM

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

Answers 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.

May 16, 2019 04:04 AM

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