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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.## 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

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 Q_{p} of p-adic numbers.

- Discrete spaces.
- The rational numbers.
- The irrational numbers.
- The p-adic numbers. More generally, profinite groups are totally disconnected.
- The Cantor set.
- The Baire space.
- The Sorgenfrey line.
- Zero dimensional T
_{1}spaces. - Stone spaces.
- The Knaster-Kuratowski fan provides an example of a totally disconnected space, such that the addition of a single point produces a connected space.

- Subspaces, products, and coproducts of totally disconnected spaces are totally disconnected.
- Totally disconnected spaces are T
_{1}spaces, since points are closed. - Continuous images of totally disconnected spaces are not necessarily totally disconnected, in fact, every compact metric space is a continuous image of the Cantor set.
- A locally compact hausdorff space is zero-dimensional if and only if it is totally disconnected.
- Every totally disconnected compact metric space is homeomorphic to a subset of a countable product of discrete spaces.

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Last updated on Wednesday June 18, 2008 at 22:53:55 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday June 18, 2008 at 22:53:55 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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