Prove that the Stone-Cech compactification of a discrete space is such that the closure of every open set is open

Question

Let $X$ be a discrete space and $\beta (X)$ its Stone-Cech compactification. We have to prove that for every open set $U$ of $\beta (X)$, $\overline{U}$ is open in $\beta (X)$. So far the only idea I have is to use the extension of some continuous function from $X$ into the discrete space $\{0,1\}$ but I'm not sure how to use it in this case. Any help would be appreciated.