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  "body": "\n## Summary\n\nAn atomless Boolean algebra is a [[Boolean algebra|0XADJKM699]] with no [[atoms|TTVCTW6TNC]]: below every nonzero element there is a strictly smaller nonzero element.\nOne example is the algebra of clopen subsets of the Cantor space.\nOthers are the [[free Boolean algebra|1MF0TVHDZD]] on infinitely many generators and the [[regular open algebra|17S8AM75AD]] of $\\R$.\nAny two countable atomless Boolean algebras with $0 \\ne 1$ are isomorphic, so there is essentially one such algebra.\nAtomless algebras are the opposite extreme to the [[complete atomic Boolean algebras|P5MA4CDRNA]], which are power sets.\n\n## Explanations\n\n### L1 human\n\nIn the subsets of a set, every nonempty set contains a single point, and one cannot split a point further.\nIn an atomless Boolean algebra there are no such smallest pieces: every nonzero element can be split again into two nonzero parts.\nThink of the finite unions of intervals in $[0, 1)$: every such nonempty set can be cut into two smaller nonempty ones.\n",
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