Arens-Fort Space is not Extremally Disconnected

Theorem
Let $T = \left({S, \tau}\right)$ be the Arens-Fort space.

Then $T$ is not extremally disconnected.

Proof
Let use the notation $S_m(V)$ for the set $S_m$ of $V\subseteq \Z_+\times Z_+$ defined in the article of arens-fort space.

Let $U$ be the set $U=\{(n,m):\exists k$ such that $m=2k \}-\{(0,0)\}$, then from the definition of Arens-Fort space $U$ is open since $(0,0)\not\in U$.

$U$ is not closed since $\complement(U)$ is not open: $(0,0)\in \complement(U)$ and $S_m(U)$ is infinite for any $m\in\N$.

From the definition of closure $U^-=U\cup \{(0,0)\}$ because $U\cup \{(0,0)\}$ is closed, since $(0,0)\not\in\complement(U\cup \{(0,0)\})$.

$U\cup \{(0,0)\}$ is not open because $(0,0)\in U\cup \{(0,0)\}$ and $S_m(U\cup\{(0,0)\})$ is infinite for any $m\in\N$.

Thus $U$ is a counterexample to show that Arens-Fort space is not  extremally disconnected.