I couldn't think anything about this question from a book by Bass:
Does there exist a $σ$-algebra which has countably many elements, but not finitely many?
May I get a hint please?
Thanks!
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Serkan Yaray
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2If it contains a countably infinite family of nonempty pairwise disjoint sets, you know that is contains at least $2^{\aleph_0}$ elements. – Daniel Fischer Nov 11 '14 at 18:32
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See here – Petite Etincelle Nov 11 '14 at 18:38