TITLE: dense subspace of $\beta \Bbb N \times \beta \Bbb N$
QUESTION [7 upvotes]: Let $\beta \Bbb N$ be a Čech-Stone compactification of the discrete space $\Bbb N$ and fix a point $p\in \beta \Bbb N\setminus \Bbb N$. Put $X=\Bbb N\cup \{p\}$ and $Y=\beta \Bbb N \setminus \{p\}$. I have two questions,

$Z=(X\times Y)\cup \{(p,p)\}$ is dense in $\beta\Bbb N\times \beta\Bbb N$ ?
$\{(p,p)\}$ is closed in $Z$?

REPLY [4 votes]: All are yes. 
Q1, $\mathbb N$ is dense in $\beta\mathbb N$, for $\beta\mathbb N$ is compactification of $\mathbb N$, and hence $\mathbb N \times \mathbb N$ is also dense in $\beta\mathbb N \times \beta\mathbb N$.
Q2, $\beta\mathbb N \times \beta\mathbb N$ is normal because it is Hausdorff compact, and hence $Z$ is at least Tychonoff (which we could see more). Note that the property Tychonoff is hereditary. And hence, $Z$ is $T_1$. Note that every point is closed in a $T_1$ space. So in the discrete space, every point is open and closed.