Definition:Totally Disconnected Space

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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