Arens-Fort Space is not Weakly Locally Compact

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

Then $T$ is not a weakly locally compact space.

Proof
We have that Neighborhood of Origin in Arens-Fort Space is not Compact.

So $\left({0, 0}\right)$ is a point in $S$ which is not contained in a compact neighborhood.

Hence, by definition, $T$ is not weakly locally compact.

Also see

 * Arens-Fort Space is Completely Hausdorff