On the Cantor- Bendixson derivatives of Lindelöf scattered spaces

Question

Suppose that $X$ is a Lindelöf scattered space. What can be said about the Cantor-Bendixson derivatives of $X$?

I know, for instance, if we consider $\mathsf{CB}(X)$ to be the least ordinal $\alpha$ such that $X^{(\alpha)}=\emptyset$, then:

1. If $\mathsf{CB}(X)=\alpha$ is a successor ordinal, then the cardinality of $X^{(\alpha-1)}$ must be at most countable;

2. If $\mathsf{CB}(X)$ is a limit ordinal, then it must have countable cofinality.

We easily see that these two properties does not imply that $X$ is Lindelöf. Indeed, for 2) just consider $Y$ the uncountable disjoint union of $\omega$ and for 1), just consider the disjoint union of $Y$ with $\omega+1$. That's why it's not enough to study $\mathsf{CB}(X)$, so we must take a look at the form of its Cantor-Bendixson derivatives. I came into the following property:

3. For any open set $U$ containing $X^{(\alpha+1)}$, $U^{\rm c}\cap X^{(\alpha)}$ must be countable.

What else can be said about them? Is there a study of this topic already in the literature?