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Intuitively for me, it seems as if closed sets are bounded, especially considering closed sets contain all limit points. But I know this isn't the case, because $ℝ$ is closed (and open) and is not bounded. Is this the only case of a closed set not being bounded? Can anyone provide an example that further illustrates the difference between closed and bounded?

Thank you.

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    One thing it might be handy to remember is that a closed set is the complement of an open set. So like, the complement of any open ball, for instance, is closed.2012-12-08

4 Answers 4

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In $\mathbb R^n$ every non-compact closed set is unbounded.

35

We cover each of the four possibilities below.

Closed and bounded: $[0,1]$

Closed and not bounded: $\cup_{n\in Z}[2n,2n+1]$

Bounded and not closed: $(0,1)$

Not closed and not bounded: $\cup_{n\in Z}(2n,2n+1)$

11

The integers as a subset of $\Bbb R$ are closed but not bounded.

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    Do you mean ℤ? So, would every finite set be closed and bounded?2012-12-08
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    yes this is because each singleton is closed and the finite union of closed sets is closed again2012-12-08
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    Finally finite implies bounded2012-12-08
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    @sugataAdhya I was referring to singletons in $\mathbb{R}$ not in any space2014-02-10
6

$$\{x\in\mathbb R\mid x\geq 0\}$$

Also note that there are bounded sets which are not closed, for examples $\mathbb Q\cap[0,1]$.

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    Thanks, can you please see my reply to Amr?2012-12-08
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    Yes, every finite set is closed and definitely bounded.2012-12-08
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    @AsafKaragila: Check out my comment above.2012-12-08
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    @Sugata: Real analysis is done in the real numbers with the standard topology.2012-12-08
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    @AsafKaragila: 'General-Topology' is tagged with the question. But as far as your solution is concerned it's all right.2012-12-08
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    @Sugata: I added the general topology tag. It was originally tagged only with real analysis tag.2012-12-08