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I have trouble grasping the difference between bounded, closed and compact sets. As a picture is worth a thousand words (especially for a person with a light math background), I would like to get a graphical representation of those concepts.

Definitions:

Bounded set A set having all its points lie within some fixed distance of each other. A set in $\mathbb{R}^n$ is bounded if all of the points are contained within a ball of finite radius

Closed set A set containing all its limit points. The closure of the set is equal to the set.

Compact set compactness is a property that generalizes the notion of a subset of Euclidean space being closed and bounded

Here is a figure that I took from this other question and modified:

bounded_closed_compact

my question

Can we say that the subfigures ($1$) and ($4$) of the figure are compact?

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  • $\begingroup$ Yes to the last question. Your first example is wrong. That set is closed and bounded - it's just the origin. (Perhaps that's a typo.) $\endgroup$
    – Ethan Bolker
    May 12, 2019 at 16:12
  • $\begingroup$ Thank you for your comment @EthanBolker. I removed the example and just left the definition and figure. Is everything correct now? $\endgroup$
    – ecjb
    May 12, 2019 at 16:18
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    $\begingroup$ In $\mathbb R^n$, closed and bounded means compact $\endgroup$
    – J. W. Tanner
    May 12, 2019 at 16:19
  • $\begingroup$ Thank you for your comment @J.W.Tanner. So we can indeed say that the subfigures (1) and (4) of the figure are compact. Correct? $\endgroup$
    – ecjb
    May 12, 2019 at 16:21
  • $\begingroup$ Everything is right now. The only example missing (for logical completeness) is the fourth possibility: not closed and not bounded (necessarily not compact, of course). $\endgroup$
    – Ethan Bolker
    May 12, 2019 at 16:22

1 Answer 1

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A subset of $\mathbb R^n$ (e.g., $\mathbb R^2$, in your depictions) is compact if and only if it is closed and bounded. As you showed, a subset could be closed but not bounded, or it could be bounded but not closed. It could also be neither closed nor bounded [such as $\mathbb R^2\setminus (0,0)]$. In any of those cases, it is not compact. As you alluded to, compact can be defined for topological spaces in general (every open cover has a finite subcover), but the Heine-Borel theorem states that for $\mathbb R^n$, a subset is compact if and only if it is closed and bounded.

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