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 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. 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${\displaystyle \,\cap \,\mathbb {Q} ^{\omega }}$ 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 totally disconnected space

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

References

• Willard, Stephen (2004), General topology, Dover Publications, ISBN 978-0-486-43479-7, MR 2048350 (reprint of the 1970 original, MR0264581)

See also

• Totally disconnected groups.