# 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?

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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.

o.h.
May 16, 2019 04:04 AM

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