Scattered Space is not necessarily T1

Theorem

Let $T = \struct {S, \tau}$ be a scattered topological space.

Then $T$ is not necessarily a $T_1$ (Fréchet) space.

Proof

Let $T = \struct {S, \tau}$ be a non-trivial particular point space.

From Particular Point Space is Scattered, $T$ is a scattered space.

From Non-Trivial Particular Point Topology is not $T_1$, $T$ is not a $T_1$ (Fréchet) space.

$\blacksquare$

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