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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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    $\begingroup$ $[0, \infty)$ in $\Bbb R$. $\endgroup$ – Henry T. Horton Aug 3 '12 at 18:42
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    $\begingroup$ The entire space. $\endgroup$ – Ink Aug 3 '12 at 18:44
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    $\begingroup$ $\mathbb{Z}$ in $\mathbb{R}$. $\endgroup$ – Makoto Kato Aug 3 '12 at 18:45
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Being closed means nothing but being the complement 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. What you are looking for.

I.e:, Any complement of any open ball!

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  • $\begingroup$ That's exactly the response I was looking for. Thank you. $\endgroup$ – ncRubert Aug 3 '12 at 18:45
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A simple example of a closed but unbounded set is $[0,\infty)$.

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