I'm trying to prove the following characterization of stability due to Rothmaler and Herzog:
$T$ is stable iff the restriction map $ f: S(\mathbb M) \to S(M)$ admits a continuous section, where $\mathbb M \models T$ is a monster and $M$ is an arbitrary model of $T$.
The "only if" direction follows easily from the open map theorem, I believe.
The following is my attempt at proving the "if" direction. There being such an $f$ means that there is a Boolean homomorphism $F$ from the boolean algebra of definable (large) sets in $\mathbb M$ to that of definable sets over $M$. The map $F$ induces $f$ in such a way that $f(p) = \{ \phi \in L(\mathbb M) \mid F(\phi(\mathbb M)) \in p\}$ (I'm abusing the notation here), so $f(p)$ is an heir of $p$. So I naturally tried to prove that $f(p)$ is the unique global heir of $p$, which would imply the stability. So I assume that $q$ is another global heir of $p$ with $\phi \in f(p)$ and $\neg\phi \in q$... And I'm stuck here.
What's the right strategy to prove this result of Rothmaler and Herzog's?