I'm wondering whether anyone has come across the following concept before:
Consider a first-order language $L$ and a type $p$ over a theory $T$. I say that $p$ has a well-ordered filter-base if there exists a transfinite sequence $(\phi_\alpha)_{\alpha < \lambda} \subseteq p$ (where $\lambda$ is some ordinal) such that $\{\phi_\alpha : \alpha < \lambda\}$ is a filter-base for $p$ (considered as a filter in the appropriate Boolean algebra) and $T \vdash (\phi_\beta \to \phi_\alpha)$ for all $\alpha < \beta < \lambda$. Such a type $p$ will correspond to a point in the space of types with a neighbourhood base linearly-ordered by $\subseteq$. I could call such a type linear. Of course, if the language were countable then $p$ would be countable, so I could enumerate it and take conjunctions of initial finite segments. But if $L$ were uncountable then this property becomes non-trivial.
Or more generally, what if $p$ were a p-point in the space of types; that is, for every countable subcollection $\Phi \subseteq p$, there exists a $\psi \in p$ such that $T \vdash (\psi \to \phi)$ for all $\phi \in \Phi$.
Thanks!