Is a star countable space $X$ with at most countable non-isolated points always Lindelof?
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 $X = \operatorname{St}(K,\mathscr{U})$.
Thanks ahead.
Yes. Suppose that $X$ is star-countable and has only countably many non-isolated points. Let $I=\{x\in X:x\text{ is isolated}\}$. Let $\mathscr{U}$ be an open cover of $X$; there is a countable $\mathscr{U}_0\subseteq\mathscr{U}$ such that $X\setminus I\subseteq\bigcup\mathscr{U}_0$. If $I\setminus\bigcup{U}_0$ is uncountable, then the open cover
$$\mathscr{U}_0\cup\left\{\{x\}:x\in I\setminus\bigcup\mathscr{U}_0\right\}$$
shows that $X$ is not star-countable. Thus, $I\setminus\bigcup\mathscr{U}_0$ is countable, and $\mathscr{U}$ has a countable subcover.