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I want to know if every Borel measurable set in the real line has cardinality either that of the naturals or of the reals. Of course the Continuum Hypothesis is not assumed.

It is clear that every open set has cardinality $2^{\aleph_0}$ (think of a ball of positive radius). I already feel lost when it comes to the closed sets.

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    $\begingroup$ "It might be helpful to know that all Borel measurable sets have cardinality either $\aleph_0$ or $2^{\aleph_0}$. Then a measurable set of cardinality strictly between those two must be Lebesgue but not Borel measurable. " --- from your last question. So you already know the answer. $\endgroup$ – Asaf Karagila May 23 '15 at 12:16
  • $\begingroup$ Actually that comment was more of a question than an assertion, it was the approach I was taking in solving the previous question. After you answered however I kept thinking about it. $\endgroup$ – Anguepa May 23 '15 at 12:22
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    $\begingroup$ @Asaf: Perhaps this is just a language problem, and the OP meant "It might be helpful to know whether...". $\endgroup$ – TonyK May 23 '15 at 12:27
  • $\begingroup$ @Tony: Perhaps. $\endgroup$ – Asaf Karagila May 23 '15 at 12:40
  • $\begingroup$ @AsafKaragila I did mean that, sorry for the misunderstanding. $\endgroup$ – Anguepa May 23 '15 at 13:18
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It is not a trivial theorem, and it certainly needs the axiom of choice (well, after a certain number of unions and complements have been applied anyway). So the proof is not very obvious.

First we show that given a Polish space $X$ (separable and completely metrizable space), then every uncountable closed set has size continuum, this is done by essentially constructing a copy of the Cantor set inside the closed set.

Next we show that given any Borel set $A$, there is a topology extending the given Polish topology such that:

  1. $A$ is clopen in the new topology.
  2. The new topology is Polish.
  3. The Borel sets are the same Borel sets.

If $A$ is uncountable, then it has to have size $2^{\aleph_0}$ since it must have a copy of the Cantor set inside of it.

You can find the proof in Kechris "Classical Descriptive Set Theory" on pp. 82-83.

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