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Let $\mathscr{B}$ be the set of Borel sets on $R$, how to prove that $\overline{\overline{\mathscr{B}}}=\aleph$? That is , the cardinality of it is continuum?

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marked as duplicate by Nate Eldredge, Mark Fantini, Travis, PhoemueX, Najib Idrissi Sep 27 '14 at 8:01

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    $\begingroup$ Hint: What's the cardinality of the set of open sets of $\mathbb{R}$? $\endgroup$ – rfauffar Sep 27 '14 at 0:54
  • $\begingroup$ We can prove it this way, for example $\endgroup$ – Omnomnomnom Sep 27 '14 at 1:13
  • $\begingroup$ Or, if you know that the cardinality of the open sets is $c$, find a way of indexing the borel sets by $\Bbb R^{\Bbb N}$ $\endgroup$ – Omnomnomnom Sep 27 '14 at 1:38
  • $\begingroup$ Hi. Omnomnomnom. Thanks for your answer. Yes. I can prove that the cardinality of the set of open sets of $R$ is $c$. Would you like to specific the way of indexing the borel sets by $R^N$? By the way, I do not like the transfinite induction. $\endgroup$ – user177634 Sep 27 '14 at 2:06

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