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Definition 1. A topological space $\langle X, \tau \rangle$ is meager if $X = \bigcup_{n \in \omega}A_n$, where each $A_n$ is nowhere dense in $\langle X, \tau \rangle$.

Definition 2. A topological space is of the second category if it is non-meager.

Definition 3. A topological space is perfect if it does not contain isolated points.

I have the following question about the second category property of uncountable subspaces of the real line:

Question. Is it provable in ZFC (is it consistent with ZFC) that every perfect uncountable subspace of the real line contains a perfect uncountable subspace that is of the second category?

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    $\begingroup$ @BjørnKjos-Hanssen This space is of the second category because it is complete metric space $\endgroup$
    – Smolin Vlad
    Commented Jun 13, 2020 at 8:18
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    $\begingroup$ It is certainly not provable in ZFC, since under Martin's Axiom, every subset of $\mathbb R$ of size less than the continuum is meager. $\endgroup$
    – Anonymous
    Commented Jun 13, 2020 at 13:16
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    $\begingroup$ Yes, under MA every subset of $\mathbb R$ of size less than the continuum is meager in itself. $\endgroup$
    – Anonymous
    Commented Jun 13, 2020 at 13:44
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    $\begingroup$ @AlessandroCodenotti, it is not clear why is it true that either set contains a Cantor subspace or is a Bernstein set, because by definition in your link, set is a Bernstein set iff it and its complement intersect with every Cantor space, i.e. it is not enough does not contain a cantor subspace in order to be a Bernstein set $\endgroup$
    – Smolin Vlad
    Commented Jun 13, 2020 at 14:14
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    $\begingroup$ Yes, I should have said every subset of size less than the continuum without isolated points. I think that this is in Miller's article in the Handbook of Set-Theoretic Topology. $\endgroup$
    – Anonymous
    Commented Jun 13, 2020 at 14:22

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