(Non-Euclidean) Compactness Compactness in Euclidean Space
The only definition of compact set that ever made sense to me was the intro calculus one:

A set is called compact if it is closed and bounded.                                                                         (*)

Compactness in Non-Euclidean Space
In contrast with (*), the "every open  cover  has finite sub cover" definition seems to be somewhat elusive and not conceptually clear to me.

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*What is the logic behind more general definition(s) of compactness?

*How is it related to the "compact = closed + bounded " one?


EDIT:

*

*More specifically, Given a set, what does the fact that  its "every open cover has a finite sub cover" has to do with the essence of compactness, which in my mind still boils down to the closeness and boundless (plus maybe some extra properties in non-Euclidean case)?



I would be happy to learn about your thoughts on the notion  on compactness.

Extra:

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*What are the other generalizations of the concept of compactness?

*What is the motivation/logic/necessity behind them?

*How is compactness of a set related to compactness of a function/operator/mapping?

 A: Like many, I first met compactness as an undergrad, and between the definition we 'intuit' in $\mathbb{R}^n$ and the topological definition, I completely understand your frustration. The 'essence' is a difficult one to tease out at first, but keep in mind what we are interested in when we study spaces in topology: the underlying structure of their 'open' sets, i.e. topological elements. This is how we re-define an equivalent definition of continuity in a first year course of point-set.
If I just approached you with the open pre-image definition of continuity, would it in any way conjure the same intuitive picture of the $\epsilon-\delta$ definition we all loved and hated in Calculus?
So compactness is another way we express topological objects relative to structure and how continuous maps behave when dealing with this entities. The meat and potatoes here is the following: with an arbitrary set $X$, the topology may well be an infinitely huge, uncountable collection of open sets. How then, are we supposed to categorize its subspaces? To generate an equivalence with the tangible picture of 'Euclidean' compactness, much like most textbooks do when first using the open pre-image definition of continuity, think about what 'closed' and 'bounded' might mean in topological terms in the general cases?
Bounded here is quite literal from the definition, as any compact set is 'covered' (read 'bounded' if it helps here!) by at most finitely many sets. This means a set is either 'small' enough (or, conversely, large enough relative to a coarse enough topology) such that we can manage to characterize it finitely using the topological entities we love so much (open sets). From the perspective of 'closedness', look at our particularly nice and well-behaved Hausdorff spaces: a compact set in such a space is necessarily closed. Every closed subspace of a compact space is closed, etc...to look at it conceptually and ditch rigor for a moment, the 'closedness' keeps other open sets out of the mix, and since we only want finitely many to intersect and cover our space, it is quite nice (and convenient) that often infinitely many do not! In a word, compactness keeps a subspace 'manageable', and in the process, gives rise to a number of other nice, manageable properties.
Now, we can throw intuition out the window and prove that there are spaces like $\mathbb{R}$ that, under the finite complement topology, bear the condition that every subspace is compact, but the real utility of compactness comes about when we further our own topological studies for more advanced properties, like countability axioms, separation axioms, compactifications (and their utility), covering spaces, homotopy groups, etc...again, compactness keeps it manageable, which is nice considering how abstract and unmanageable topology can seem as a subject sometimes. I hope this helps!
A: Second definition is more general than the first, and can be applied to any topological space.  In $\Bbb R^n$ the two definitions are equivalent, but this is not true for other spaces. 
The finite cover definition abstracts many of the properties of closed and bounded set in $\Bbb R^n$, so that they can be applied to other settings. Some of these properties include: 


*

*the fact that any infinite set of points in a closed bounded set has a limit point, and 

*the image of a closed bounded set under a continuous map between Euclidean spaces is also closed and bounded.

A: To me the intuitive notion of compactness is sequential compactness, i.e. that every sequence has a convergent subsequence, and the open cover definition is only motivated by the Heine-Borel theorem. Of course the open cover definition is more general, in that it makes sense in any topological space, but understanding what it really means takes quite a bit more effort and experience. Personally, I don't think just the Heine-Borel theorem was enough for me to really understand. I had to think about the extreme value theorem, the Heine-Cantor theorem, etc. to really get a feel for it.
With all of that in mind, completeness (which follows from closedness if the underlying space is complete) and boundedness are both necessary conditions for compactness in any metric space. Both of these facts have easy proofs, which I'll sketch here. In a non-complete set, you can choose a sequence which converges to a point on the boundary which is not in the set, and this sequence cannot have a convergent subsequence. In a non-bounded set, you can choose a sequence whose distances from some fixed point monotonically blow up, and this sequence will have no convergent subsequence.
In $\mathbb{R}^n$, these two conditions together are sufficient. This result, which is called the Bolzano-Weierstrass theorem, is significantly harder to prove than the previous results, and uses properties which are specific to $\mathbb{R}^n$. Many important results in real analysis (e.g. the Arzela-Ascoli theorem) amount to identifying sufficient conditions for compactness in spaces other than $\mathbb{R}^n$ (including certain nonmetrizable spaces).
This is related to compact operators in a simple fashion: a linear operator is compact if it takes bounded subsets of its domain to precompact subsets of its codomain. (A set is precompact if its closure is compact.)
