# 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 Qp of p-adic numbers.

## Definition

A topological space X is totally disconnected if the connected components in X are the one-point sets.

## Examples

The following are examples of totally disconnected spaces:

## 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. Endow $X/{\sim}$ with the quotient topology, i.e. the coarsest 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, then it uniquely factors into $f=\breve{f}\circ m$ where $\breve{f}:(X/\sim)\rightarrow Y$ is continuous.