TITLE: What does a compact set look like?
QUESTION [6 upvotes]: So for a writing assignment in one of my classes we are asked to discuss and prove some basic results about compact sets in general topological spaces. I like proving these things, but they dont help me understand what a compact( locally compact, paracompact,...) set in a topological space "looks like." That said I'm asking for some examples of compact (locally compact, ...) sets in a variety of topological spaces. I'm also interested in some explanation of what extra "benefits" are brought about by singling out these compact sets. For example the p-adic numbers are locally compact and locally compact things (abelian groups to be precise) are a good setting in which to carry out fourier analyis.

REPLY [6 votes]: You could try reading Terence Tao's notes on the subject; I found them very informative.
As for examples, here is an example and a non-example which I think are informative.  The example is that, by Tychonoff's theorem, $[0, 1]^I$ is compact for any index set $I$.  The non-example is that the closed unit ball in any infinite-dimensional Banach space, say $\ell_1(\mathbb{Z})$, is not compact.  
Edit:  Here is another perspective from which to think about compact Hausdorff spaces (and LCH spaces).  A compact Hausdorff space $X$ is completely determined by the ring of continuous functions $C(X, \mathbb{R})$; this is a standard exercise which is worked out, for example, here.  One gets the points of $X$ as the maximal ideals of $C(X, \mathbb{R})$ and the topology as the initial topology making every function in $C(X, \mathbb{R})$ continuous.  Slightly generalized, this is the commutative Gelfand-Naimark theorem, and it says that studying commutative C*-algebras with unit is the same thing as studing compact Hausdorff spaces.  Removing the requirement that we have a unit is the same thing as studying locally compact Hausdorff spaces.
So one can write down compact Hausdorff spaces by writing down commutative C*-algebras with unit.  One choice is to take the space $C_b(X, \mathbb{R})$, the space of bounded continuous functions on an arbitrary topological space $X$, with the sup norm.  If $X$ is completely regular Hausdorff, this gives the Stone-Cech compactification of $X$.  (Which is hopeless to think about in general; even for $X = \mathbb{N}$ it's very weird, as the Wikipedia article describes.)