Arens-Fort Space is Scattered

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Theorem

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


Then $T$ is a scattered space.


Proof

Let $H \subseteq T$ such that $H \ne \left\{{\left({0, 0}\right)}\right\}$.

Then $\exists x \in H: x \ne \left({0, 0}\right)$.

From Clopen Points in Arens-Fort Space, every point of $T$ apart from $\left({0, 0}\right)$ is open in $T$.

So $\left\{{x}\right\}$ is an open set of $T$.

So $H \cap \left\{{x}\right\} = \left\{{x}\right\}$ and so $x$ is isolated in $H$.

Thus $H$ contains at least one point which is isolated in $H$.


On the other hand, suppose $H = \left\{{\left({0, 0}\right)}\right\}$.

From Singleton Point is Isolated, $\left({0, 0}\right)$ is an isolated point in $\left\{{\left({0, 0}\right)}\right\} = H$.

So again $H$ contains at least one point which is isolated in $H$.


So for all $H \subseteq S$ we have that $H$ contains at least one point which is isolated in $H$.

Hence the result, by definition of scattered space.

$\blacksquare$


Sources