Totally disconnected space

In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons 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 integers. Another example, playing a key role in algebraic number theory, is the field Template:Math of p-adic numbers.

Definition

A topological space <math>X</math> is totally disconnected if the connected components in <math>X</math> are the one-point sets.Template:SfnTemplate:Sfn Analogously, a topological space <math>X</math> is totally path-disconnected if all path-components in <math>X</math> are the one-point sets.

Another closely related notion is that of a totally separated space, i.e. a space where quasicomponents are singletons. That is, a topological space <math>X</math> is totally separated if for every <math>x\in X</math>, the intersection of all clopen neighborhoods of <math>x</math> is the singleton <math>\{x\}</math>. Equivalently, for each pair of distinct points <math>x, y\in X</math>, there is a pair of disjoint open neighborhoods <math>U, V</math> of <math>x, y</math> such that <math>X= U\sqcup V</math>.

Every totally separated space is evidently totally disconnected but the converse is false even for metric spaces. For instance, take <math>X</math> to be the Cantor's teepee, which is the Knaster–Kuratowski fan with the apex removed. Then <math>X</math> 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<ref>Template:Cite book</ref> totally disconnected spaces are sometimes called hereditarily disconnected,Template:Sfn while the terminology totally disconnected is used for totally separated spaces.Template:Sfn

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<math>\, \cap \, \mathbb{Q}^{\omega}</math> 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

• Subspaces, products, and coproducts of totally disconnected spaces are totally disconnected.
• Totally disconnected spaces are T₁ spaces, since singletons 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

Let <math>X</math> be an arbitrary topological space. Let <math>x\sim y</math> if and only if <math>y\in \mathrm{conn}(x)</math> (where <math>\mathrm{conn}(x)</math> denotes the largest connected subset containing <math>x</math>). This is obviously an equivalence relation whose equivalence classes are the connected components of <math>X</math>. Endow <math>X/{\sim}</math> with the quotient topology, i.e. the finest topology making the map <math>m:x\mapsto \mathrm{conn}(x)</math> continuous. With a little bit of effort we can see that <math>X/{\sim}</math> 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 holds: For any totally disconnected space <math>Y</math> and any continuous map <math>f : X\rightarrow Y</math>, there exists a unique continuous map <math>\breve{f}:(X/\sim)\rightarrow Y</math> with <math>f=\breve{f}\circ m</math>.

See also

• Extremally disconnected space
• Totally disconnected group

References

• Template:Munkres Topology
• Template:Rudin Walter Functional Analysis
• Template:Citation (reprint of the 1970 original, Template:MR)
• Template:Citation