Scattered T1 Space is Totally Disconnected

Theorem

Let $T = \struct {S, \tau}$ be a scattered topological space which is also a $T_1$ (Fréchet) space.

Then $T$ is totally disconnected.

Proof

Let $T = \struct {S, \tau}$ be a scattered space which is also a $T_1$ (Fréchet) space.

We have that every Non-Trivial Connected Set in $T_1$ Space is Dense-in-itself.

As $T$ is scattered, every $H \subseteq S$ contains at least one point which is isolated in $H$.

So $H$ is not dense-in-itself and so if $H$ has more than one element it can not be connected.

As $H$ is arbitrary, it follows that $T$ is totally disconnected.

$\blacksquare$

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