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I have read that a subset of Euclidean space may be called compact if it is both closed and bounded. I was wondering what a good example of a closed but unbounded set would be?

Would a closed ball inside a sphere with an infinite radius do the trick? If that example works are there any other examples people could think of?

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$[0, \infty)$ in $\Bbb R$. – Henry T. Horton Aug 3 '12 at 18:42
The entire space. – Ink Aug 3 '12 at 18:44
$\mathbb{Z}$ in $\mathbb{R}$. – Makoto Kato Aug 3 '12 at 18:45
up vote 7 down vote accepted

Being closed means nothing but begin the completement of an open set. So take any bounded open subset $S \subset \mathbb R^n$, then $\mathbb R^n \setminus S$ is closed but not bounded, hence what you look for.

E.g: Any complement of any open ball!

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That's exactly the response I was looking for. Thank you. – ncRubert Aug 3 '12 at 18:45

simple exampe closed but unbounded $[0,\infty)$

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