How to show that remainder $\beta X \setminus \beta(X)$ is an $F_\sigma$-set in $\beta X$. The $\beta \mathbb N$ be the Stone-Čech compactification of $\mathbb N$.
I have seen in Engelking - General topology that the remainder $\beta X \setminus \beta(X)$ is an $F_\sigma$-set in $\beta X$. 
Can anyone explain why?
Thank you! 
 A: Consider $X$ (assumed to be Tychonoff, so $\beta X$ exists) to be a subset of its compactification $\beta X$. Then the remainder $\beta X \setminus X$ is an $F_\sigma$ in $\beta X$ iff $X$ is a $G_\delta$ in $\beta X$. This condition is known as $X$ being "topology complete", and for metrisable spaces it turns out to be equivalent to being completely metrisable. 
Also, it's easy to see that the remainder is closed iff $X$ is open in $\beta X$, which means exactly that $X$ is locally compact. (So note that all locally compact Tychonoff spaces are topologically complete.)
As $\mathbb{N}$ is locally compact, the remainder is thus even closed. But e.g. for $X = \mathbb{R}$, or $X = \mathbb{R}\setminus\mathbb{Q}$ we also have $F_\sigma$ remainders, while for $X = \mathbb{Q}$ we do not, as the latter is not topology complete (not completely metrisable). 
So it depends on $X$. 
A: The remainder $β\mathbb{N} \setminus \mathbb{N}$ is even closed in $β\mathbb{N}$ since any point of $\mathbb{N}$ is isolated.
