Scattered Space is T0
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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.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness: Disconnectedness