Discrete Space is Scattered

Theorem
Let $T = \left({S, \vartheta}\right)$ be a topological space where $\vartheta$ is the discrete topology on $S$.

Then $T$ is a scattered space.

Proof
We have that All Points in Discrete Space are Isolated.

So, by definition, no closed set $H \subseteq S$ of $T$ such that $H \ne \varnothing$ is dense-in-itself.

So, again, by definiton, $T$ is scattered.