Isolated Points in Arens-Fort Space

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Theorem

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

Let $q \in S: q \ne \left({0, 0}\right)$.


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


Proof

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

In particular, $\left\{{q}\right\}$ is open in $T$.

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

$\blacksquare$


Sources