# Totally disconnected space

(Redirected from Totally disconnected)

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 Qp 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:

## Constructing a totally 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 $f={\breve {f}}\circ m$ .