Either-Or Topology is Scattered

Theorem
Let $T = \left({X, \tau}\right)$ be the either-or space.

Then $T$ is a scattered space.

Proof
Let $x \in T, x \ne 0$.

By definition of either-or space, $\left\{{x}\right\}$ is open in $T$.

So $\left\{{x}\right\}$ is an open set of $x$ containing only $x$, and so $x$ is isolated.

So a subset of $H \subseteq X$ contains isolated points and so by definition is not dense-in-itself.

If $H = \left\{{0}\right\}$, then from Singleton Set is Not Dense-in-itself $H$ is again not dense-in-itself.

So by definition $T$ is a scattered space.