---
tag: 4E4RXKV9W3
kind: statement
role: theorem
name: van Dantzig's theorem
msc: [22D05, 22A05]
uses: [AHV05ZBCYW, AW3J9AQSA0, QYCFH68RD1, 0R6EWFKRCF, 0ZCY0Z80B6]
about: [AW3J9AQSA0]
sources:
  - url: https://en.wikipedia.org/wiki/Totally_disconnected_group
---

## Statement

Let $G$ be a Hausdorff [[locally compact group|AHV05ZBCYW]] that is [[totally disconnected|AW3J9AQSA0]].
Then every [[neighbourhood|0ZCY0Z80B6]] of the identity $e$ contains a [[compact|0R6EWFKRCF]] [[open subgroup|QYCFH68RD1]] of $G$.

## Remarks

The proof has two steps.
A locally compact Hausdorff totally disconnected space has a basis of compact open sets, so the given neighbourhood contains a compact open set $V \ni e$.
By compactness of $V$ there is a symmetric open neighbourhood $W$ of $e$ with $V W \subseteq V$.
Then all products of elements of $W$ lie in $V$, so the subgroup generated by $W$ is open and lies in $V$.
An open subgroup is closed, so this subgroup is compact.

Example: in the $p$-adic numbers $\Q_p$, the subgroups $p^k \Z_p$ with $k \ge 0$ are compact and open, and they shrink to $0$.
In $\mathrm{GL}_n(\Q_p)$, the congruence subgroups $I + p^k \mathrm{M}_n(\Z_p)$ with $k \ge 1$ play the same role.

Local compactness is necessary.
The group $\Q$ with the topology of $\R$ is totally disconnected and not locally compact.
An open subgroup of it contains an interval around $0$, so it is all of $\Q$, which is not compact.

## Explanations

### L1 human

In a totally disconnected group, points cannot be joined by any connected piece.
Van Dantzig's theorem says that, when the group is also locally compact, there are arbitrarily small compact subgroups that are also open.
They play the role of small balls around the identity.
For example, in the $p$-adic numbers the balls $p^k \Z_p$ are themselves subgroups.

### L2 human

The theorem is the starting point of the structure theory of totally disconnected locally compact groups.
It shows that such a group is a union of cosets of a compact open subgroup $U$, with a discrete coset space $G/U$.
A totally disconnected compact group is profinite, so these groups are locally profinite.
Compact open subgroups are used to define [[Cayley–Abels graphs|QWG66JE1JE]] and the [[Hecke algebras|6H3WCCHZB5]] of $p$-adic groups.