# Discrete Space is Totally Disconnected

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## Theorem

Let $T = \struct {S, \tau}$ be a topological space where $\tau$ is the discrete topology on $S$.

Then $T$ is totally disconnected.

## Proof

Follows from:

- Discrete Space is Extremally Disconnected
- Extremally Disconnected Space is Totally Separated
- Totally Separated Space is Totally Disconnected

$\blacksquare$

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $6.5$: Components