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