Definitions

Disconnected space

Totally disconnected space

In topology and related branches of mathematics, a totally disconnected space is a topological space which is maximally disconnected, in the sense that it has no non-trivial connected subsets. In every topological space the empty set and the one-point sets are connected; in a totally disconnected space these are the only connected subsets.

An important example of a totally disconnected space is the Cantor set. Another example, playing a key role in algebraic number theory, is the field Qp of p-adic numbers.

Definition

A topological space X is totally disconnected if the connected components in X are the one-point sets.

Examples

The following are examples of totally disconnected spaces:

Properties

References

See also

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