I've been trying to show that the Morley rank of a formula is equal to the Cantor-Bendixson (CB) rank of the clopen subset associated to that formula in the Stone space of types over a monster model.
More precisely, I'm trying to show that if we have $(\phi_i)_{i<\omega}$ such that the CB rank of $[\phi_i]$ is at least $\alpha$ and that $[\phi_i] \subseteq [\phi]$, then the CB rank of $[\phi]$ is at least $\alpha + 1$.
Ideally, I could show that the CB derivative of $[\phi]$ includes $[\phi_i]$ for some $i$, which would imply the claim above. I don't know why this is true, since $\phi_i$ might have an isolated point of $\phi$.
How can I prove this induction step?
(For the other direction, I would imagine that I can just repeatedly cut $[\phi]$ into halves, which I think can be done assuming the type space is big enough (infinite?). But I don't have a clear idea here, either.)
Addendum: I realized that one might have to adopt another definition of CB ranks than the one used in general topology. According to this alternate definition, a set whose CB derivatives stabilizes with a nonempty set must have CB rank $\infty$.