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I am able to see why the closed nowhere dense subsets of $\mathbb R$ are equinumerous with $\mathbb R$: Every closed nowhere dense subset of $\mathbb R$ is the boundary of an open set (namely its compliment). $\mathbb R$ has $2^{\aleph_0}$ many open sets, so there are at most $2^{\aleph_0}$ many closed nowhere dense subsets of $\mathbb R$. Every Cantor set is a closed nowhere dense subset of $\mathbb R$, and there are $2^{\aleph_0}$ many different Cantor sets, so there are precisely $2^{\aleph_0}$ many closed nowhere dense subsets of $\mathbb R$.

My next thought was to take closures, but this is not injective. What am I missing?

Just a hint, please!

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    $\begingroup$ Why do you think there are continuum many nowhere dense sets on the line? $\endgroup$
    – Asaf Karagila
    Commented Nov 9, 2018 at 16:14
  • $\begingroup$ @AsafKaragila Because I misread a proof in Just and Weese! I will edit the nature of the question to adapt, since I am now curious. $\endgroup$
    – TiddSchmod
    Commented Nov 9, 2018 at 16:17
  • $\begingroup$ If $A$ is nowhere dense, surely any subset of $A$ is nowhere dense as well. Do you have any other requirements on the sets? $\endgroup$ Commented Nov 9, 2018 at 16:17
  • $\begingroup$ @AndrésE.Caicedo Hmm. That's a good point. Nevermind, I no longer believe that my question has a positive answer. $\endgroup$
    – TiddSchmod
    Commented Nov 9, 2018 at 16:18
  • $\begingroup$ I am very flattered that you confuse me and @Andrés, unfortunately for me, we are not the same person... $\endgroup$
    – Asaf Karagila
    Commented Nov 9, 2018 at 16:22

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As pointed out in the comments, there are more than continuum many nowhere dense subsets of $\mathbb R$. In particular, the Cantor set $C$ has the same size as the reals, and any subset of $C$ is nowhere dense, so there are at least $|2^{C}| = 2^{2^{\aleph_0}} > 2^{\aleph_0}$ such sets.

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    $\begingroup$ (Also, there are only $2^{2^{\aleph_0}}$ sets of reals, so the number you got is as large as possible.) $\endgroup$ Commented Nov 9, 2018 at 16:58

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