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Ex: how to prove the sphere

$$\left\{ (x,y,z)∈ℝ^3∣x^2+y^2+z^2=1 \right\}$$ is a closed set??

I tried to use the definition of the closed set,but it did not work out for me.

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1  
Which definition of closed set are you using? –  Carlos Eugenio Thompson Pinzón Oct 16 '13 at 18:44
    
It may be easier to show that the complement is open. –  David Mitra Oct 16 '13 at 18:44
    
I tried use the the complements is open,but i cant really show that X^2+y^2+z^2>1 is open. –  Dan0210 Oct 16 '13 at 18:49
    
You might also find Is a sphere a closed set useful. –  J. W. Perry Oct 16 '13 at 18:50

2 Answers 2

up vote 1 down vote accepted

My hint would be: show that the "inside" is open and the "outside" is open.

To add a bit more details: Let $\tilde{x}$ denote an arbitrary point in $\mathrm{Outside}=\{(x,y,z) \mid x^2 + y^2+z^2 > 1\}.$ Thinking about it geometrically, the ball centered at $\tilde{x}$ of radius $|\tilde{x}| -1$ should be a subset of $\mathrm{Outside}$. So, you should try showing this.

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Hint: Use the fact that inverse images of closed sets through continuous functions are closed.

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