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 $\big\vert \overline{F} \big\vert \leq 2^\omega $.
This is a step of a proof of Corollary 1 in this paper, the complete result is that $\vert X \vert\leq 2^\omega$ and 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.
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 $X$ with $\vert \gamma \vert\leq 2^\omega$ there exists a countable subfamily $\eta\subset \gamma $ that covers $Y$.
For each $x\in\operatorname{cl}F$ let $\mathscr{B}_x$ be a countable base at $x$, and for each $B\in\mathscr{B}_x$ fix $p_B\in B\cap F$. Let $F_x=\{p_B:B\in\mathscr{B}_x\}$. Let
$$\varphi:\operatorname{cl}F\to\left[[F]^{\le\omega}\right]^{\le\omega}:x\mapsto\{B\cap F_x:B\in\mathscr{B}_x\}\;,$$
where $[A]^{\le\kappa}$ is the family of subsets of $A$ of cardinality at most $\kappa$. Because $X$ is Hausdorff, we have
$$\{x\}=\bigcap\{\operatorname{cl}(B\cap F_x):B\in\mathscr{B}_x\}\;,$$
so $x$ can be uniquely recovered from $\varphi(x)$, and $\varphi$ is therefore injective. But then
$$|\operatorname{cl}F|\le\left|\left[[F]^{\le\omega}\right]^{\le\omega}\right|=\left(|F|^{\omega}\right)^{\omega}\le2^\omega\;.$$