---
tag: G852FHZ3W1
kind: concept
name: Bohr compactification
msc: [22C05, 43A60, 22B05]
sources:
  - url: https://en.wikipedia.org/wiki/Bohr_compactification
same_as:
  wikidata: Q4938217
  wikipedia: Bohr compactification
---

## Summary

The Bohr compactification of a topological group $G$ is the largest compact Hausdorff group into which $G$ maps continuously.
Every continuous homomorphism from $G$ to a compact Hausdorff group factors through it in a unique way.
For a locally compact abelian group it is the dual of the dual group made discrete.
The continuous functions on it correspond to the almost periodic functions on $G$.

## Explanations

### L1 human

Many groups, like the real line, are not compact, but compact groups are easier to analyse.
The Bohr compactification is the best compact group that "sees" $G$: it keeps exactly the information that compact groups can detect.
For the real line, it keeps all the waves $x \mapsto e^{i\xi x}$ but forgets the topology of the frequencies $\xi$.
The resulting compact group is huge, but the continuous functions on it correspond exactly to the almost periodic functions studied by Harald Bohr, such as $\cos x + \cos(\sqrt{2}x)$.