TITLE: Stone-Čech Compactification of the Natural numbers
QUESTION [9 upvotes]: I am trying to prove that if $U$ is contained in the Stone-Čech Compactification of the natural number ($\beta N$) that the closure of $U$ is open. 
I have a really hard time with even understanding what the Stone-Čech Compactification is, so I can't even start this problem. Any suggestions would be greatly appreciated.
Thank you!

REPLY [3 votes]: Like those who have answered previously (GEdgar and Paul), I'll assume you intended $U$ to be a subset of $\mathbb N$, even though the question assumes only $U\subseteq\beta\mathbb N$.  (If one takes the question literally, the result is just false.)  Once you see how $\beta\mathbb N$ is constructed using ultrafilters, it will be easy to check that, if $\mathbb  N$ is partitioned into two pieces, then the closures of those pieces constitute a partition of $\beta\mathbb N$.  In particular, $(\beta\mathbb N)-\overline U=\overline{N-U}$ is closed, and therefore $\overline U$ is open.