---
tag: NFEEGWJFCK
kind: concept
name: Atomless Boolean algebra
msc: ["06E05"]
specializes: [0XADJKM699]
sources:
  - url: https://en.wikipedia.org/wiki/Atom_(order_theory)
---

## Summary

An atomless Boolean algebra is a [[Boolean algebra|0XADJKM699]] with no [[atoms|TTVCTW6TNC]]: below every nonzero element there is a strictly smaller nonzero element.
One example is the algebra of clopen subsets of the Cantor space.
Others are the [[free Boolean algebra|1MF0TVHDZD]] on infinitely many generators and the [[regular open algebra|17S8AM75AD]] of $\R$.
Any two countable atomless Boolean algebras with $0 \ne 1$ are isomorphic, so there is essentially one such algebra.
Atomless algebras are the opposite extreme to the [[complete atomic Boolean algebras|P5MA4CDRNA]], which are power sets.

## Explanations

### L1 human

In the subsets of a set, every nonempty set contains a single point, and one cannot split a point further.
In an atomless Boolean algebra there are no such smallest pieces: every nonzero element can be split again into two nonzero parts.
Think of the finite unions of intervals in $[0, 1)$: every such nonempty set can be cut into two smaller nonempty ones.