What should be the intuition when working with compactness?

I have a question that may be regarded by many as duplicate since there's a similar one at MathOverflow. The point is that I think I'm not really getting the idea on compactness. I mean, in $\mathbb{R}^n$ the compact sets are those that are closed and bounded, however the guy who answered this question and had his answer accepted says that compactness is some analogue of finiteness.

That's the first problem: In my intuitive view of finiteness, only boundedness would suffice to say that a certain subset of $\mathbb{R}^n$ is in some sense "finite". On the other hand there's the other definition of compactness (in terms of covers) which is the one I really need to work with and I cannot see how that definition implies this intuition on finiteness.

Also, I feel it's pretty strange the covers people use when they want to deal with compact sets. To prove a set is compact I know they must show that for every open cover there's a finite subcover; the problem is that I can't see intuitively how one could show this for every cover. Also when trying to disprove compactness the books I've read start presenting strange covers that I would have never thought about. I think my real problem is that I didn't yet get the intuition on compactness.

So, what intuition should we have about compact sets in general and how should we really put this definition to use?

Can someone provide some reference that shows how to understand the process of proving (and disproving) compactness?

Solution 1:

The following story may or may not be helpful. Suppose you live in a world where there are two types of animals: Foos, which are red and short, and Bars, which are blue and tall. Naturally, in your language, the word for Foo over time has come to refer to things which are red and short, and the word for Bar over time has come to refer to things which are blue and tall. (Your language doesn't have separate words for red, short, blue, and tall.)

One day a friend of yours tells you excitedly that he has discovered a new animal. "What is it like?" you ask him. He says, "well, it's sort of Foo, but..."

The reason he says it's sort of Foo is that it's short. However, it's not red. But your language doesn't yet have a word for "short," so he has to introduce a new word - maybe "compact"...

The situation with compactness is sort of like the above. It turns out that finiteness, which you think of as one concept (in the same way that you think of "Foo" as one concept above), is really two concepts: discreteness and compactness. You've never seen these concepts separated before, though. When people say that compactness is like finiteness, they mean that compactness captures part of what it means to be finite in the same way that shortness captures part of what it means to be Foo.

But in some sense you've never encountered the notion of compactness by itself before, isolated from the notion of discreteness (in the same way that above you've never encountered the notion of shortness by itself before, isolated from the notion of redness). This is just a new concept and you will to some extent just have to deal with it on its own terms until you get comfortable with it.

Solution 2:

You may read various descriptions and consequences of compactness here. But be aware that compactness is a very subtle finiteness concept. The definitive codification of this concept is a fundamental achievement of $20^{\,\rm th}$ century mathematics.

On the intuitive level, a space is a large set $X$ where some notion of nearness or neighborhood is established. A space $X$ is compact, if you cannot slip away within $X$ without being caught. To be a little more precise: Assume that for each point $x\in X$ a guard placed at $x$ could survey a certain, maybe small, neighborhood of $x$. If $X$ is compact then you can do with finitely many (suitably chosen) guards.