Is the following true? Given a first countable Hausdorff space $X$ and a dense subset $Y\subset X$ which is initially $2^\omega$-initially Lindelöf in $X$. Let $F$ be a subset of $Y$ such that $\vert F\vert\leq 2^{\omega }$, then $\vert \overline{F} \vert \leq 2^\omega $.

I actually want to prove something stronger, namely that $Y$ is Lindelöf in $X$. The proof goes by transfinite induction on $\alpha<\aleph_1$, one of the steps require the above statement to be true, so maybe some of the hypothesis above are not needed for that particular statement. Also by Lindelöf in $X$ I mean that for any open cover $\gamma$ of $X$ there exists a countable subfamily $\eta\subset\gamma $ that covers $Y$. By $2^\omega-$initially Lindelöf I mean that for every open cover $\gamma$ of $\vert \gamma \vert\leq 2^\omega$ there exists a countable subfamily $\eta\subset \gamma $ that covers $Y$