Arens-Fort Space is Zero Dimensional

Theorem

Let $T = \struct {S, \tau}$ denote the Arens-Fort space.

Then $T$ is zero dimensional.

Proof

Let $q \in S$ such that $q \ne \tuple {0, 0}$.

Then from Clopen Points in Arens-Fort Space, $\set q$ is clopen.

So $\forall q \in S, q \ne \tuple {0, 0}: \set {\set q}$ is a local basis for $q$.

If we take the neighborhoods of $\tuple {0, 0}$ that are open we get that they are also closed from Neighborhood of Origin of Arens-Fort Space is Closed.

The union of the local basis forms a basis for the topology.

This basis is formed with clopen sets.

So, by definition, $T$ is a zero dimensional space.

$\blacksquare$

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: $8$