Arens-Fort Space is Paracompact

Theorem

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

Then $T$ is a paracompact space.

Proof 1

Let $\mathcal C$ be any open cover of $T$.

Let $H \in \mathcal C$ be any open set which contains $\left({0, 0}\right)$.

For all $s \in S$ such that $s \ne \left({0, 0}\right)$, we have that $\left\{{s}\right\}$ is open in $T$ by definition of the Arens-Fort space.

So the open cover of $T$ which consists of $H$ together with all the open sets $\left\{{s}\right\}$, where $s \in S \setminus H$ is a refinement of $T$ which is locally finite.

Hence the result, by definition of paracompact space.

$\blacksquare$

Proof 2

We have that:

The Arens-Fort Space is Completely Normal.
The Arens-Fort Space is Lindelöf.

From Sequence of Implications of Separation Axioms, it follows that $T$ is a $T_3$ space.

The result follows from Lindelöf $T_3$ Space is Paracompact.

$\blacksquare$