# 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.

$\blacksquare$