Definition:Totally Disconnected Space

Definition

A topological space $T = \struct {S, \tau}$ is a totally disconnected space if and only if all components of $T$ are singletons.

That is, $T$ is a totally disconnected space if and only if it contains no non-degenerate connected sets.

Also defined as

Because of Totally Disconnected but Connected Set must be Singleton, the definition for totally disconnected space is applied by some authors to a topological space containing at least $2$ elements.

Also see

• Totally Disconnected but Connected Set must be Singleton

• Definition:Totally Pathwise Disconnected Space

• Results about totally disconnected spaces can be found here.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $6.5$: Components
• 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
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): totally disconnected