# Visual representation of difference between closed, bounded and compact sets

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:

my question

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

• 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.) May 12 '19 at 16:12
• Thank you for your comment @EthanBolker. I removed the example and just left the definition and figure. Is everything correct now?
– ecjb
May 12 '19 at 16:18
• In $\mathbb R^n$, closed and bounded means compact May 12 '19 at 16:19
• 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?
– ecjb
May 12 '19 at 16:21
• 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). May 12 '19 at 16:22

## 1 Answer

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.