Discrete Space is Totally Disconnected

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