Does closed imply bounded?

Question

Definitions:
1. A set $S$ in $\mathbb{R}^m$ is bounded if there exists a number $B$ such that $\mathbf{||x||}\leq B$ for all $\mathbf{x}\in S$, that is , if $S$ is contained in some ball in $\mathbb{R}^m$.
2. A set in $\mathbb{R}^m$ is closed if, whenever $\{\mathbf{x}_n\}_{n=1}^{\infty}$ is convergent sequence completely contained in $S$, its limit is also contained in $S.$
3. A set $S$ in $\mathbb{R}^m$ is compact if and only if it is both closed and bounded.

Does closedness not imply boundedness in general? If so, why does a compact set need to be both closed as well as bounded?

The whole space is closed, certainly not bounded. The set of points with integer coordinates is closed, not bounded. — André Nicolas, Nov 24 '13 at 17:38

score 22 · Accepted Answer · answered Nov 24 '13 at 17:37

22

The set $\{\,(x,y)\in\mathbb R^2\mid xy=1\,\}$ is closed but not bounded.

Even simpler, $\mathbb R^n$ itself is closed (but not bounded).

answered Nov 24 '13 at 17:37

Hagen von Eitzen

374,180

score 3 · Answer 2 · answered Nov 24 '13 at 17:37

3

$\mathbb{R}^m$ itself is a closed set. is it bounded?

But in case of Compact sets, they are closed as well as bounded in $\mathbb{R}^n$.

answered Nov 24 '13 at 17:37

GA316

4,324
27
50

Does closed imply bounded?

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