---
tag: AW3J9AQSA0
kind: concept
name: Totally disconnected group
msc: [22A05, 54H11]
specializes: [92700QAB5Y]
sources:
  - url: https://en.wikipedia.org/wiki/Totally_disconnected_group
same_as:
  wikidata: Q7828252
  wikipedia: Totally disconnected group
---

## Summary

A totally disconnected group is a [[topological group|92700QAB5Y]] whose underlying space is [[totally disconnected|0RJSBR9GW1]].
Equivalently, its identity component is the trivial subgroup.
Profinite groups, $p$-adic groups and discrete groups are examples.
For a totally disconnected locally compact group, van Dantzig's theorem gives a base of compact open subgroups at the identity.
This is the starting point of the structure theory of such groups.

## Explanations

### L1 human

A totally disconnected group has no connected pieces except single points.
Any two distinct elements can be separated, so the group looks like dust rather than like a curve or a surface.
The $p$-adic integers are the main example.
Two $p$-adic integers are close when their difference is divisible by a high power of $p$, and the space falls apart into small separate balls.
Every group with the discrete topology is also totally disconnected.