$T_2$ + Hereditarily Lindelöf implies countable pseudocharacter

Question

Recently, I have been wondering about the relation between separation axioms, countable pseudocharacter, and other properties that a topological space may have. One of the questions that I had was if $T_2$ + having a countable $k$-network (or having a countable network) implies countable pseudocharacter. In my previous question (see here), where I was looking for an example of a $T_1$ space that has a countable $k$-network and that doesn't have countable pseudocharacter, this question has also been been made by @Jakobian in the comments. I also answered in the comments, but I wanted to make a new question for this result.

The following answer is the proof that I have come up with, which requires the space to be $T_2$, but just hereditarily Lindelöf.

Answer 1

Let $X$ be a $T_2$ and hereditarily Lindelöf space.

Given a point $x$ of $X$. $X \setminus \{x\}$ is Lindelöf. And, for each $y \in X \setminus \{x\}$, as $X$ is $T_2$, there exists an open neighborhood $A_y$ of $y$ in $X$ such that $x$ is not in the closure of $A_y$. As $\{A_y\}_{y \in X \setminus \{x\}}$ is an open cover of $X \setminus \{x\}$ in $X$, it has a countable subcover, and $X \setminus \{x\}$ is equal to the union of the closures of the elements of this subcover. Therefore, $X \setminus \{x\}$ is a $F_{\sigma}$ set. As a result, $\{x\}$ is a $G_{\delta}$ set.

We conclude that $X$ has countable pseudocharacter.