Is there a star countable, semi-stratifiable space $X$ with $|X|> \mathfrak c$?

## 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 **star countable** if whenever $\mathscr{U}$ is an open cover of $X$, there is a countable subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$, where $\operatorname{St}(K,\mathscr{U}) = \bigcup \{ U \in \mathscr{U} : K \cap U \neq \emptyset \}$.