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How to prove that algebra generated by a countable set is countable. Hint enough. I know that algebra generated by any set $A$ is of the form $\cup_{i=1}^{m}\cap_{j=1}^{n_i} A_{ij}$ where $A_{ij}$ or $A_{ij}^c$ is in $A$.

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  • Can you provide the form of $\sigma(\mathscr{A})$, where $\mathscr{A}$ is an algebra. How does it differ from yours? – Mike Brown Mar 13 '15 at 19:18

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HINT: Every set in the algebra has a "generating sequence" which is a finite sequence including the sets and indices over which we union and intersect. Note that if $A$ is a countable set then set of finite sequences from $A$ is countable.

Asaf Karagila
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  • Could you link some explanation/proof of the "generating sequence" idea? –  Aug 11 '19 at 19:24
  • David, it's easiest if you take some specific example to understand the idea. The point is that to be in the algebra means that you're some union of some sets of their complements, which were previously generated. In the Linked to your right there's a question that is essentially a follow-up to this one which might also be helpful. – Asaf Karagila Aug 11 '19 at 21:43