You might be interested by this article by Terry Tao.
"Compactness and Compactification"
Which contains a really great discussion of compactness, applications of compactness, and what compactness means. But my favorite part of the article is the end, since it is the only part that contained a statement which really surprised me. Which I will quote directly, not feigning to describe something better than Terry Tao:
Another use of compactifications is to allow one to rigorously view one type of
mathematical object as a limit of others. For instance, one can view a straight
line in the plane as the limit of increasingly large circles, by describing a suitable
compactification of the space of circles which includes lines; this perspective allows
us to deduce certain theorems about lines from analogous theorems about circles,
and conversely to deduce certain theorems about very large circles from theorems
about lines. In a rather different area of mathematics, the Dirac delta function is
not, strictly speaking, a function, but exists in a certain (local) compactification
of spaces of functions, such as spaces of measures or distributions. Thus one can
view the Dirac delta function as a limit (in a suitably weak topology) of classical
functions, which can be very useful for manipulating that function. One can also use
compactifications to view the continuous as the limit of the discrete; for instance,
it is possible to compactify the sequence Z/2Z, Z/3Z, Z/4Z, etc. of cyclic groups,
so that their limit is the circle group T = R/Z. These simple examples can be
generalised into much more sophisticated examples of compactifications (and to the
closely related concept of completions), which have many applications in geometry,
analysis, and algebra.