---
tag: CZG6Y6EAHP
kind: definition
name: Totally disconnected group
msc: [22A05, 54H11]
defines: AW3J9AQSA0
uses: [92700QAB5Y, 0RJSBR9GW1, V6TJAFZ4MD]
sources:
  - url: https://en.wikipedia.org/wiki/Totally_disconnected_group
---

## Definition

A *totally disconnected group* is a [[topological group|92700QAB5Y]] $G$ whose underlying topological space is [[totally disconnected|0RJSBR9GW1]].
That is, the only connected subsets of $G$ are the empty set and the one-point sets.

## Remarks

Example: the [[p-adic integers|V6TJAFZ4MD]] $\Z_p$ under addition.
Each subgroup $p^n \Z_p$ is open and closed, and two distinct elements $x, y$ lie in different cosets of $p^n \Z_p$ for $n$ large, so no connected set contains both.
Non-example: $(\R, +)$ with the usual topology is connected, so it is not totally disconnected.
Equivalently, $G$ is totally disconnected if and only if its identity component is $\set{e}$, because each connected component is a coset of the identity component.