Existence of Connected Non-T1 Scattered Space

Theorem

There exists at least one example of a connected topological space which is not a $T_1$ (Fréchet) space, which is also a scattered space.

Proof

Let $T$ be a divisor space.

From Divisor Space is Connected, $T$ is a connected space.

From Divisor Space is not $T_1$, $T$ is not a $T_1$ (Fréchet) space.

From Divisor Space is Scattered, $T$ is a scattered space.

Hence the result.

$\blacksquare$

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