# Definition:Totally Disconnected Space

Jump to navigation
Jump to search

## Contents

## Definition

A topological space $T = \left({S, \tau}\right)$ 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

- Results about
**totally disconnected spaces**can be found here.

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 4$: Disconnectedness - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $6.5$: Components