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  "body": "\n## Definition\n\nLet $B$ be a Boolean algebra.\nAn [[atom|TTVCTW6TNC]] of $B$ is an element $a \\ne 0$ such that there is no $b$ with $0 \u003c b \u003c a$.\nThe algebra $B$ is a *complete atomic Boolean algebra* if:\n\n1. $B$ is a [[complete Boolean algebra|75BPV95AAS]].\n2. $B$ is [[atomic|5SE8EMSDNY]]: for every $b \\ne 0$ in $B$ there is an atom $a$ with $a \\le b$.\n\n## Remarks\n\nIf $B$ is complete and atomic, with set of atoms $A$, then $b \\mapsto \\set{a \\in A : a \\le b}$ is an isomorphism from $B$ onto the power set of $A$.\nSo the complete atomic Boolean algebras are, up to isomorphism, the power set algebras.\n\nExample: the power set of a set $X$.\nIt is complete, and every nonempty $Y \\subseteq X$ contains a singleton $\\set{x}$, which is an atom.\n\nNon-example: the algebra of [[regular open sets|833XAA2Q65]] of $\\R$ is complete but has no atom at all.\nEvery nonempty regular open set contains an open interval, which contains two disjoint smaller open intervals, and these are regular open and strictly smaller.\nSo clause 2 fails.\n\nIn Mathlib the notion is `CompleteAtomicBooleanAlgebra`.\n",
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