[Math] Stone-Čech Compactification of the Natural numbers

Question

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!

Answer 1

Lemma 1 (see the Engelking's book Corollary 3.6.5): For every open and closed subset $A$ of a Tychonoff space $X$ the closure $\overline{A}$ of $A$ in $\beta X$ is open and closed.

Proof: Notice that $U \cap N$ is not empty, since $U$ is open and $N$ is dense in $\beta X$. Moreover, $U \cap N$ is open and closed in $N$, then we apply the Lemma 1 to conclude that $\overline{U\cap N}$ is open in $\beta X$. Finally, note that $\overline{U}=\overline{U\cap N}$, since $N$ is dense in $\beta N$. Thus we conclude that $\overline{U}$ is open in $\beta X$.