We have $X$ an $F$-space, $Y$ a subspace of $X$ whose complement is of the first category. Prove that $Y=X$. This problem have a hint: need to show that $Y $ intersect $x+Y$ $\forall x \in X$, but i still don't know how do it. Thanks in advace.
It is clear that for every $x \in X$, $(x+Y) \cap Y \neq \emptyset$. Indeed, let us suppose that there exists $x \in X$ s.t. $x+Y \subset X \setminus Y$. Then $x+Y$ is of the first category, hence - since addition is a linear homeomorphism - $Y$ is of the first category. This is absurd, because we would have $X=Y \cup (X \setminus Y)$ so $X$ of the first category, absurd by Baire ($X$ is a complete metric space).
Now we have that for every $x \in X$ there exists $y \in Y \cap (x+Y)$, i.e. there exists $y'$ s.t. $y=x+y'$, hence $x=y-y' \in Y$ because $Y$ is a linear subspace. Then $X \subseteq Y$, which implies $X=Y$.