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?
Best 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.