Closed set on Euclidean space that is not compact

Question

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 03 '12 at 18:42 — Henry T. Horton, Aug 03 '12 at 18:42
$\mathbb{Z}$ in $\mathbb{R}$. – Makoto Kato Aug 03 '12 at 18:45 — Makoto Kato, Aug 03 '12 at 18:45

score 34 · Accepted Answer · edited Dec 29 '19 at 03:06

34

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!

edited Dec 29 '19 at 03:06

Learning stats by example

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answered Aug 03 '12 at 18:44

Dario

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That's exactly the response I was looking for. Thank you. – ncRubert Aug 03 '12 at 18:45

score 10 · Answer 2 · edited Jun 12 '18 at 02:07

10

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

edited Jun 12 '18 at 02:07

Anonymous Pi

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answered Aug 04 '12 at 06:53

Myshkin

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Closed set on Euclidean space that is not compact

2 Answers2