Totally disconnected space

In topology and related branches of mathematics, a totally disconnected space is a topological space that 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. Analogously, a topological space X is totally path-disconnected if all path-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, 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∩ $\mathbb{Q}^{\omega}$ 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 disconnected space

Let $X$ be an arbitrary topological space. Let $x\sim y$ if and only if $y\in \mathrm{conn}(x)$ (where $\mathrm{conn}(x)$ 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 $m:x\mapsto \mathrm{conn}(x)$ continuous. With a little bit of effort we can see that $X/{\sim}$ is totally disconnected. We also have the following universal property: if $f : X\rightarrow Y$ a continuous map to a totally disconnected space $Y$ , then there exists a unique continuous map $\breve{f}:(X/\sim)\rightarrow Y$ with $f=\breve{f}\circ m$ .

References

Willard, Stephen (2004), General topology, Dover Publications, ISBN 978-0-486-43479-7, MR 2048350 (reprint of the 1970 original, MR 0264581)