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 &quot;finite&quot;. 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?
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.
In my opinion, I think it's okay that you treat compactness as a clever trick as you said. But as it is an abstract concept, I think it's recommended to take the concept as itself, not comparing to real examples.
Maybe the concept of compactness is motivated from Heine-Borel property, but even without a motivation, I think the definition of compactness itself is very natural.
There are many branches of mathematics and each branch has its main vocabularies. For example, in number theory, one's main vocabulary is an &quot;integer&quot;. Just like this, &quot;openness&quot; is a main vocabulary in general topology. (The question of asking why openness is a main vocabulary is a completely different question and this is well-answered in wikipedia.) So when you do general topology, you play with open sets. And definitely it's hard to make an argument on infinite open objects, but compactness ease this so that when you play with compactness you can only consider finite objects (for each compact set).
If you take concepts in this way, definitions of Lindelöf space and paracompact space and etc are very natural just like compactness. I'm quite sure no one has a clear &quot;real life example&quot; of paracompactness but almost all of people here think the definition is very natural.