Scattered Space is T0

Theorem
Let $T = \struct {S, \tau}$ 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 Equivalence of Definitions of $T_0$ Space, there exist $x, y \in S$ such that $x$ and $y$ are both limit points of each other.

So by definition of isolated point, neither $x$ nor $y$ are isolated in $\set {x, y}$.

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

So $T$ is not scattered.

Hence the result by Rule of Transposition.