Vol. 40, No. 3, 1972

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Cardinality of k-complete Boolean algebras

W. Wistar (William) Comfort and Anthony Wood Hager

Vol. 40 (1972), No. 3, 541–545

An infinite complete Boolean algebra satisfies |B|0 = |B| (where || denotes cardinality). This is a theorem of R. S. Pierce, derived in consequence of his general decomposition theorem [9]. It is here shown (directly) that |B|0 = |B| for B merely countably complete; this has the corollary (actually, equivalent) that if A is an algebra of measurable functions modulo null functions, and D is a subset of A which is dense in the uniform topology, then |D| = |A|. The relation |B|f = |B| for f-complete Boolean algebras B is considered; the main result is a structure theorem for the nontrivial counterexamples (which are shown to exist abundantly).

Mathematical Subject Classification
Primary: 06A40
Received: 2 November 1970
Published: 1 March 1972
W. Wistar (William) Comfort
Anthony Wood Hager