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 $Q_{p}$ 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.

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 ) 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$$\, \cap \, \mathbb{Q}^{\omega}$$ 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 quotient space of any given 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.

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