# Totally Disconnected Space is T1

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## Theorem

Let $T = \left({S, \tau}\right)$ be a topological space which is totally disconnected.

Then $T$ is a $T_1$ (Fréchet) space.

## Proof

Let $T = \left({S, \tau}\right)$ be totally disconnected.

Then as its components are singletons, it follows that each of its points is closed.

From Equivalence of Definitions of $T_1$ Space, it follows that $T$ is a $T_1$ (Fréchet) space.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 4$: Disconnectedness