# 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.

## DefinitionEdit

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.

## ExamplesEdit

The following are examples of totally disconnected spaces:

## Constructing a disconnected spaceEdit

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 ${\displaystyle f={\breve {f}}\circ m}$ .