Questions about an example

Question

Recently, I met an example.

I have two questions about the example: Why the author said, because $z \notin A$, then $z$ is not in the closure in $\beta \mathbb{R}$ of $A \cap (\beta \mathbb{R} \setminus \mathbb{R})$?

And It seems that the point $z$ is a $P$-point is useless in this example. Am I right?

(A point of a space is a $P$-point if the point belongs to the interior of any $G_\delta$-set which contains this point.)

Thanks ahead:)

Answer 1

The fact that $z$ is a $P$-point is used to answer your first question. $A\cap(\beta\Bbb R\setminus\Bbb R)$ is a countable set in a $T_1$-space, so it’s an $F_\sigma$-set; $z$ is a $P$-point in $\beta\Bbb R\setminus\Bbb R$, so it’s not in the closure of any $F_\sigma$-set in $\beta\Bbb R\setminus\Bbb R$.