Clopen 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 $\left\{{q}\right\}$ is both open and closed in $T$.


$\left\{{\left({0, 0}\right)}\right\}$ itself is closed, but not open.


Proof

We have that $\left\{{q}\right\}$ is finite so $\complement_S \left({\left\{{q}\right\}}\right)$ is cofinite.

So $\complement_S \left({\left\{{q}\right\}}\right)$ is open and so $\left\{{q}\right\}$ is closed.


Then we have that $\left({0, 0}\right) \notin \left\{{q}\right\}$ so $\left\{{q}\right\}$ is open.


However, $\left({0, 0}\right) \notin \complement_S \left({\left\{{\left({0, 0}\right)}\right\}}\right)$ and $\complement_S \left({\left\{{\left({0, 0}\right)}\right\}}\right)$ is clearly not open in $T$.

But as $\left\{{\left({0, 0}\right)}\right\}$ is finite so $\complement_S \left({\left\{{\left({0, 0}\right)}\right\}}\right)$ is cofinite.

Hence, as for $\left\{{q}\right\}$, we have that $\left\{{\left({0, 0}\right)}\right\}$ is closed.

$\blacksquare$


Sources