Compact Sets in Projective Space Consider the projective space ${\mathbb P}^{n}_{k}$ with field $k$.  We can naturally give this the Zariski topology.
Question:  What are the (proper) compact sets in this space?
Motivation: I wanted nice examples of spaces and their corresponding compact sets; usually my spaces are Hausdorff and my go-to topology for non-Hausdorff-ness is the Zariski topology.  I wasn't really able to find any proper compact sets which makes me think I'm doing something wrong here.

You are in for a big surprise, james:  every subset of $\mathbb P^n_k$ is quasi-compact.
This is true more generally for any noetherian space, a space in which  every  decreasing  sequence of  closed sets is stationary.
However: the compact subsets of $\mathbb P^n_k$ are the finite sets of points such that no point is in the closure of another. 
Reminder
 A topological space $X$ is quasi-compact if from every open cover of  $X$ a finite cover can be extracted. A compact space is a Hausdorff quasi-compact space.    
Bibliography Bourbaki, Commutative Algebra, Chapter II, §4,2. 
