In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons (and the empty set) as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) 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, which is homeomorphic to the set of p-adic integerss. Another example, playing a key role in algebraic number theory, is the field of p-adic numberss.

Definition

is totally disconnected if the connected components are the one-point sets. is totally path-disconnected if all path-components are the one-point sets.

Another closely related notion is that of a totally separated space, i.e. a space where quasicomponents is totally separated, the intersection of all clopen neighborhoods.

Every totally separated space is evidently totally disconnected but the converse is false even for metric spaces to be the Cantor's teepee, which is the Knaster–Kuratowski fan is totally disconnected but its quasicomponents are not singletons. For locally compact Hausdorff spaces the two notions (totally disconnected and totally separated) are equivalent.

Confusingly, in the literature totally disconnected spaces are sometimes called hereditarily disconnected, while the terminology totally disconnected is used for totally separated spaces.

Examples

The following are examples of totally disconnected spaces:

• Discrete spaces

• The rational numbers

• The irrational numbers

• The p-adic numbers; more generally, all profinite groups are totally disconnected.

• The Cantor set and the Cantor space

• The Baire space

• The Sorgenfrey line

• Every Hausdorff space of small inductive dimension 0 is totally disconnected

• The Erdős space ℓ2 is a totally disconnected Hausdorff space that does not have small inductive dimension 0.

• Extremally disconnected Hausdorff spaces

• Stone spaces

• The Knaster–Kuratowski fan provides an example of a connected space, such that the removal of a single point produces a totally disconnected space.

Properties

• Subspacess, products, and coproducts of totally disconnected spaces are totally disconnected.

• Totally disconnected spaces are T1 spaces, since connected components 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 has small inductive dimension 0 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 in general not true that every open set in a totally disconnected space is also closed.

• It is in general not true that the closure of every open set in a totally disconnected space is open, i.e. not every totally disconnected Hausdorff space is extremally disconnected.

Constructing a totally disconnected quotient space of any given space

). This is obviously an equivalence relation with the quotient topology, i.e. the finest topology is totally disconnected.

In fact this space is not only some totally disconnected quotient but in a certain sense the biggest: The following universal property, there exists a unique.

See also

• Extremally disconnected space

• Totally disconnected group

Citations

References

• (reprint of the 1970 original, )