Scattered Space is T0

Theorem
Let $T = \left({S, \vartheta}\right)$ be a scattered topological space.

Then $T$ is also a $T_0$ (Kolmogorov) space.

Proof
Suppose $T$ is not a $T_0$ (Kolmogorov) space.

From Equivalent Definitions for $T_0$ Space, there exist $x, y \in S$ such that $x$ and $y$ are both limit points of each other.

So from Point is Isolated Iff Not a Limit Point, neither $x$ nor $y$ are isolated in $\left\{{x, y}\right\}$.

Thus we have found a subset $\left\{{x, y}\right\} \subseteq T$ such that $\left\{{x, y}\right\}$ is by definition dense-in-itself.

So $T$ is not scattered.

Hence the result by Rule of Transposition.