TITLE: Could *I* have come up with the definition of Compactness (and Connectedness)?
QUESTION [28 upvotes]: Ok, buckle up for a rather long question. I've spent a large portion of today learning about compactness, stemming mainly from this wikipedia article about point-set topology. The article mentions three main things: continuity, connectedness, and compactness. I will address each in turn, but my question is mainly about the last one.
Continuity: At least having gone through high school calculus and basic university analysis, I think people have a great intuitive understanding of continuity (and also differentiability I guess): smooth = yay!, jagged = fine-ish, holes/jumps = really bad :(. The wiki article describes this as "taking nearby points to nearby points", which I understand well enough that I think given a decade or something I could eventually have come up with the formal epsilon-delta definition for continuity:
$$\forall \varepsilon >0, \exists \delta, \text{ s.t. } |x-x_{0}|<\delta \Rightarrow |f(x)-f(x_{0})|<\varepsilon $$
Connectedness: Similarly, I think people have a great intuition for connectedness (at least path connectedness), which the wiki article summarizes nicely as "sets that cannot be divided into two pieces that are far apart". Again, I think that given a decade or two I could have gone in the right direction at least for the formal definition of connectedness:
a set that can't be represented as the union of two or more disjoint non-empty open subsets.

First (Minor) Question: could we have a useful definition for connectedness being: "a set that can't be represented as the union of two or more disjoint non-empty closed subsets"?

Similarly, I feel like I could reasonably develop definitions for open sets (based on intuitions from number lines and basic high-school algebra/set theory), and completeness (basically the Least Upper Bound axiom/Cauchy sequences). However, there's one thing missing. Compactness. Never in a million years do I think I could have thought up of the definition that a set that can be "can be covered by finitely many sets of arbitrarily small size." I've looked at these five sites and some links therein:

Why is compactness so important?
What should be the intuition when working with compactness?
https://www.math.ucla.edu/~tao/preprints/compactness.pdf
https://www.reddit.com/r/math/comments/47h6hg/what_really_is_a_compact_set/
https://arxiv.org/abs/1006.4131

but none of them so far has really clicked with me. Many people emphasized that it's a generalized version of finiteness with "fat blurry points", and I also understand that by the Heine-Borel Theorem compactness is equivalent to "closed and bounded" in Euclidean space, but those two things seem so far apart that it just seems like a black-magic coincidence that they describe the same phenomenon.

How would you motivate and explain the definition and concept of compactness to your students in such a way that they feel like they could have come up with it themselves, naturally, given a decade or two?
If you start with "it's a generalized version of finiteness" it seems like a complete and utter coincidence that it happens to be equivalent to "closed and bounded". I mean of all the possible "generalized finiteness formulations", how did ours get it right?
If you start with "it's just another way of saying closed and bounded", then students will feel that it's just more arbitrary confusion redefining things they already know (namely that of closed-ness and boundedness); furthermore, even if they did accept this explanation, they would have never figured out on their own that "every open cover has a finite subcover $\iff$ closed and bounded". "Finite subcover" just seems so out-of-left-field.
And finally if you go the sequential compactness route (referring to Tao's paper here) students will just say "ahh yes, the Bolzano-Weierstrass Theorem! Why need to give it a new name of compactness?"

Maybe I've missed something in my searches, but I hope this question isn't just a badly rehashed old question. I don't think my question is answered in the "Pedagogical History of Compactness" mainly because I don't want the convoluted history, but rather the simplest motivation and explanation based off modern curriculum and notation.
-> Also, thank you to those who've commented/left an answer. I hope that this page and all the differing pedagogical interpretations presented will serve as a relatively complete and comprehensive guide for beginners learning compactness in the future. Please upvote answers you think are particularly insightful; as a novice, I would appreciate some expert judgement on the explanatory power of these answers.

REPLY [2 votes]: Let me give an example drawn from real life: I have led high-school students to conjecture and prove the compactness of the unit interval.
The context was in a development of elementary measure theory. After developing the definition of measure, a proposed measure was the Jordan content: given a set $A \subseteq \mathbb{R}$, 
$$J(A) = \inf \left\{ \sum_{i=1}^n (b_i - a_i) : A \subseteq \bigcup_{i=1}^n (a_i,b_i)\right\},$$
where the infimum is over all such coverings of $A$ by intervals.
Of course, this is not a measure, since it cannot satisfy countable additivity. This leads to the definition of outer measure:
$$ m^\ast(A) = \inf \left\{ \sum_{i=1}^\infty (b_i - a_i) : A \subseteq \bigcup_{i=1}^\infty (a_i,b_i)\right\}.$$
Now one would like to show this is a translation-invariant measure on (measurable) subsets of $\mathbb{R}$. Further, it should have the property
$$m^\ast([0,1]) = 1,$$
which was true of Jordan content! (This can be proved by induction on the size of a cover.) But there are infinite covers of the interval: can they have total length less than 1? When students try and fail to construct such a cover, they are trying to construct infinite covers with no subcover. They eventually will conjecture compactness in the following form:

If $[0,1] \subseteq \cup_{i=1}^\infty (a_i,b_i)$, then there is an $N$ such that $[0,1] \subseteq \cup_{i=1}^N (a_i,b_i)$. 

In this context, the burden of motivation shifts from compactness to countable additivity--why is that a reasonable condition for a notion of length to satisfy? But this can also be done: we believe $\mathbb{Q}$ has less length than $\mathbb{R}\setminus \mathbb{Q}$, although neither of their characteristic functions are Riemann-integrable.
In sum, compactness can be discovered when there is a need for it.