Dgdrxrt

I'm still working on meagerness, and what I post is something that I had achieved yesterday but today I don't remember how I did that.
At pp. $48$, Kechris (Classical Descriptive Set Theory) defines, for a topological space $X$, a meager set $Asubseteq X$ in an open subset $Usubseteq X$ to be a subset s.t. $Acap U$ is meager in $X$.

And he says that it is equivalent to require that $Acap U$ be meager in $U$ w.r.t. the relative topology.

It's clear that meagerness in $X$ implies meagerness in $U$ (essentially by monotonicity of inclusion), but I have some problems to prove the converse.
Indeed, meager in $U$ means that there is $F_n$ s.t. $$emptyset=mathrmInt_U[mathrmCl_U(F_ncap U)]=(overlineF_ncap U)^°cap U,$$
from which I deduce that either $(overlineF_ncap U)^°=emptyset$ or they are disjoint.

Well, if I can prove that disjointness is impossibile, I'm done, but I'm not sure that this holds in general.

Notation: $overline(.)$ is the closure operator on $X$, while $(.)^°$ is the interior operator on $X$.

If $E subset U$ and if the closure of $E$ in $U$ has empty interior in $U$ then the closure of $E$ in $X$ has empty interior in $X$. This is because If $V$ is a non-empty open set in $X$ contained in the $X-$ closure of $E$ then $Vcap U$ cannot be empty: $V$ has to intersect $E$ (since any point of $V$ is in the $X-$ closure of $E$, so the neighborhood $V$ of that point has to intersect $E$) and $E subset U$ so $E$ intersects $U$. Thus $V cap U$ is a non-empty open set in $U$ contained in the closure of $E$ in $U$, contradicting the hypothesis. Taking $E=Acap U$ shows that if $Acap U$ is meager in $X$ the it is meager in $U$.