Isolated Points in Arens-Fort Space

Theorem

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

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

Then $q$ is an isolated point of $T$.

Proof

If $q \ne \tuple {0, 0}$ then from Clopen Points in Arens-Fort Space we have that $\set q$ is both closed and open in $T$.

In particular, $\set q$ is open in $T$.

The result follows from Point in Topological Space is Open iff Isolated.

$\blacksquare$

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: $8$