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.
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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
- Extremally disconnected Hausdorff 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.
- It is not true that every open set is also closed.
- It is not true that the closure of every open set is open, i.e. not every totally disconnected Hausdorff space is extremally disconnected.
References
- Willard, Stephen (2004), General topology, Dover Publications, MR2048350, ISBN 978-0-486-43479-7
See also
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