Arens-Fort Space is not First-Countable

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

Then $T$ is not a first-countable space.

Proof
Assume that $T$ is first-countable.

Then there exists a countable local base $B_0 = \left\{{U_i}\right\}_{i=1}^\infty$ for $\left({0, 0}\right)$.

By the definition of neighborhood of $\left({0, 0}\right)$, for each $U_i \in B_0$ there is a point $\left({n_i, m_i}\right) \in U_i$ such that $n_i > i$ and $m_i > i$.

We build now a new set $E = X \setminus \left\{{\left({n_i, m_i}\right)}\right\}_{i=1}^\infty$.

This new set is a neighborhood of $\left({0, 0}\right)$ by definition.

However, there is not $U_i \subseteq E$ because $\left({n_i, m_i}\right) \notin E$.

From this contradiction it follows that $T$ cannot be first-countable.