---
tag: 75BPV95AAS
kind: concept
name: Complete Boolean algebra
msc: ["06E10"]
specializes: [0XADJKM699]
sources:
  - url: https://en.wikipedia.org/wiki/Complete_Boolean_algebra
same_as:
  wikidata: Q5156447
  wikipedia: Complete Boolean algebra
---

## Summary

A complete Boolean algebra is a [[Boolean algebra|0XADJKM699]] in which every subset has a supremum, so that it is also a [[complete lattice|D7A0ZGZT5S]].
Power sets are complete, and so are the algebras of [[regular open sets|833XAA2Q65]] of topological spaces.
Every Boolean algebra embeds densely into a complete one, its completion.
Complete Boolean algebras are the basic tool of Boolean-valued models and of forcing in set theory.

## Explanations

### L1 human

In a Boolean algebra one can always take "or" of two or of finitely many elements.
In a complete Boolean algebra one can also take "or" of infinitely many elements at once, and the result is still in the algebra.
The subsets of a set form such an algebra: the "or" of any family of subsets is their union.
The finite and cofinite subsets of $\N$ do not.
The set of even numbers is a union of finite sets, but it is neither finite nor cofinite, and the family has no least upper bound in the algebra.