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I have to prove that a subset $A \subset F$ of meager set $F$ is also a meager set.

From the definition of meager set $F = \bigcup_{n=1}^{\infty}F_n$, where $F_n$ is a nowhere dense set.

As I know we have to show that $A$ has also similiar form, but the question is how do we find these nowhere dense sets, that sum to the whole $A$?

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Notice that

$A=F\cap A=\bigcup_{n=1}^\infty F_n\cap A$.

Is $A_n:=F_n\cap A$ a nowhere dense set?

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  • $\begingroup$ I get the idea, but technically I don't know how to justify, that such intersection is also nowhere dense. Any hints? $\endgroup$
    – janusz
    Oct 7 '16 at 11:07
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    $\begingroup$ @janusz Hint: a subset of a nowhere dense set is nowhere dense. $\endgroup$
    – bof
    Oct 7 '16 at 11:58

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