Arens-Fort Space is not Weakly Locally Compact

Theorem

Let $T = \struct {S, \tau}$ 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 $\tuple {0, 0}$ is a point in $S$ which is not contained in a compact neighborhood.

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

$\blacksquare$

Also see

• Arens-Fort Space is Completely Hausdorff

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $26$. Arens-Fort Space: $4$