Scattered Space is T0

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Theorem

Let $T = \left({S, \tau}\right)$ 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 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 $\left\{{x, y}\right\}$.

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.

$\blacksquare$


Sources