PlanetMath: totally disconnected space

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totally disconnected space

(Definition)

A topological space is called totally disconnected if all connected components of are singletons.

Examples of totally disconnected spaces include:

Every discrete space,
Cantor set,
the set of rational numbers $\mathbb{Q}$ endowed with the subspace topology induced by $\mathbb{Q}\hookrightarrow \mathbb{R}$ and,
the set of irrational numbers $\mathbb{R}\setminus\mathbb{Q}$ again with the subspace topology.

"totally disconnected space" is owned by Dr_Absentius.

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Other names:

totally disconnected

Also defines:

totally disconnected space

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54D99 (General topology :: Fairly general properties :: Miscellaneous)

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