Scattered T1 Space is Totally Disconnected
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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