Totally disconnected space explained

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 of p-adic numbers.

Definition

A topological space

X is totally disconnected if the connected components in X are the one-point sets. Analogously, a topological space X is totally path-disconnected if all path-components in X 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

X is totally separated if for every x\in X , the intersection of all clopen neighborhoods of x is the singleton \{x\} . Equivalently, for each pair of distinct points x, y\in X , there is a pair of disjoint open neighborhoods U, V of x, y such that X = U\sqcup V .

Every totally separated space is evidently totally disconnected but the converse is false even for metric spaces. For instance, take

X to be the Cantor's teepee, which is the Knaster–Kuratowski fan with the apex removed. Then X 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 (for instance^[1]) 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

\cap Q ^{\omega} 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

X be an arbitrary topological space. Let x\sim y if and only if y\in conn (x) (where conn (x) denotes the largest connected subset containing x ). This is obviously an equivalence relation whose equivalence classes are the connected components of X . Endow X/{\sim} with the quotient topology, i.e. the finest topology making the map m:x\mapsto conn (x) continuous. With a little bit of effort we can see that X/{\sim} 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

Y and any continuous map f : X → Y , there exists a unique continuous map \breve{f}:(X/\sim) → Y with f = \breve{f}\circ m .

See also

• Extremally disconnected space
• Totally disconnected group

References

• • • (reprint of the 1970 original,)

Notes and References

1. Book: Engelking . Ryszard . Ryszard Engelking . General Topology . Heldermann Verlag, Sigma Series in Pure Mathematics. 1989. 3-88538-006-4.