Totally 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 "Q""p" 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:
*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 T1 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.
Properties
*Subspaces, products, and coproducts of totally disconnected spaces are totally disconnected.
*Totally disconnected spaces are T1 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.
References
* | year=2004
ee also
* Totally disconnected groups.
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