# Totally disconnected space

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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 proper 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 ${\displaystyle X}$  is totally disconnected if the connected components in ${\displaystyle X}$  are the one-point sets. Analogously, a topological space ${\displaystyle X}$  is totally path-disconnected if all path-components in ${\displaystyle X}$  are the one-point sets.

Another closely related notion is that of a totally separated space, i.e. a space where quasicomponents are singletons. Equivalently, a topological space ${\displaystyle X}$  is totally separated space if and only if for every ${\displaystyle x\in X}$ , the intersection of all clopen neighborhoods of ${\displaystyle x}$  is the singleton ${\displaystyle \{x\}}$ . Equivalently, for each pair of distinct points ${\displaystyle x,y\in X}$ , there is a pair of disjoint open neighborhoods ${\displaystyle U,V}$  of ${\displaystyle x,y}$  such that ${\displaystyle X=U\sqcup V}$ .

Every totally separated space is evidently totally disconnected but the converse is false even for metric spaces. For instance, take ${\displaystyle X}$  to be the Cantor's teepee, which is the Knaster–Kuratowski fan with the apex removed. Then ${\displaystyle X}$  is totally disconnected but its quasicomponents are not singletons. For locally compact Hausdorff spaces the two notions (totally disconnected and totally separated) are equivalent.

Unfortunately in the literature (for instance [1]), 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:

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