# 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?

• $[0, \infty)$ 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

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.
A simple example of a closed but unbounded set is $[0,\infty)$.