Is every semi-stratifiable space $\omega$-monolithic?

## Definitions

A topological space $(X,\tau)$ is called **semi-stratifiable** if there exists a function $g:\omega\times X\to\tau$ such that:

for any point $x$ of $X$ holds $\{x\}=\bigcap_{n\in\omega} g(n,x)$;

for any point $x$ of $X$ and a sequence $\{x_n\}$ of $X$, if $x \in g(n,x_n)$ for each $n$, then $x_n \to x$.

A topological space $X$ is said to be **$\omega$-monolithic** if $nw(\overline{A}) \le \omega$ for any subset $A \subset X$ with $|A| \le \omega$.

$nw(X)$ denotes the cardinal function called network weight, which is minimal cardinality of a network $$nw(X)=\min\{|\mathcal N|: \mathcal N \text{ is a net for } X\}+\omega.$$

If not. What if $X$ is semi-metric space?

Note that $X$ is semi-metrisable iff $X$ is first countable and semi-stratifiable;