TITLE: Properties that better "capture" compactness.
QUESTION [7 upvotes]: I am not asking for properties equivalent to compactness, but for those that better capture the motivation for compactness, i.e. that explain why compactness is talked about so much. 
The way I see it compactness captures (in the form of sequential compactness) a property of closed and bounded subsets in an euclidean space: every sequence has a converging subsequence. Addressing this property first came Bolzano's theorem (1817) and then results on spaces of functions like the Arzelà–Ascoli theorem. Simultaneously there was the study of the continuum and proof that a continuous function defined on a closed and bounded interval is uniformly continuous (1870), which led to the modern definition of compactness (in terms of open covers), which is closely related to sequential compactness but suitable for general topology. 
My question would be the following:

If you had to write a short list of properties (say no more than five) that you'd consider the most important consequences of compactness (in any context, from general topology to a more specific setting), a sort of guideline of what compactness means, which ones would you choose?

REPLY [2 votes]: Let $f$ be lower semicontinuous and coercive, i.e., $f(x)\to +\infty$ whenever $\|x\|\to+\infty$. Then $f$ has a minimizer. 

This beautiful result, in Euclidean spaces, generalizes the Weierstrass theorem and lies at the heart of optimization: existence of minimizers! At its core is Bolzano-Weierstrass and thus compactness.
In infinite-dimensional Hilbert space, we require that $f$ be weakly lower semicontinuous, so weak sequential compactness (Eberlein-Smulyan!) is the key.