# Closed set on Euclidean space that is not compact

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. –  Vectk Aug 3 '12 at 18:44
$\mathbb{Z}$ in $\mathbb{R}$. –  Makoto Kato Aug 3 '12 at 18:45

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