To clarify for readers, this answer uses the notation of the previous version of this question and addresses some simplicity notions I've removed. $\quad$ - NS
This is just a comment, but it is too long for the comment section.
Given a "generator" $\rho$, of the topology $\tau$, as Noah defined it, one could define $\varphi: \mathcal{P}(\omega^{\omega}) \rightarrow \tau$ by $\varphi(A) = \bigcup_{f \in A} \rho(f)$. $\varphi$ is a surjective $\subseteq$-homomorphism. One could then take the quotient of $\mathcal{P}(\omega^{\omega})$ by $A \sim B$ iff $\phi(A) = \phi(B)$. We can completely recover $\tau$ (considered as a pointless topology) from this quotient together with the covering relation Noah defined. We can also recover it completely from just the $\sim$-invariant binary relation $\mathcal{R}$ on $\mathcal{P}(\omega^{\omega})$ defined by $A\mathcal{R}B$ iff $\phi(A)\supseteq \phi(B)$
This allows any topology which has a "generator" to be presented as a binary covering relation $\mathcal{R} \subseteq \mathcal{P}(\omega^{\omega}) \times \mathcal{P}(\omega^{\omega})$. So we are in the realm of higher-order descriptive set theory. It wouldn't be hard to axiomatize exactly which such $\mathcal{R}$ yield a topology, but I won't do this because I would probably leave something out by accident.
A presentation of $\tau$ in this form witnesses the $BCP_0^+$ condition if: $\exists \alpha < \Theta$ $\forall g \in \omega^{\omega}$ the upward $\subseteq$ closure of $\{A : A\mathcal{R}\{g\} \text{ and } A \text{ has Wadge degree } < \alpha \}$ is $\{A : A \mathcal{R} \{g\}\}$
A presentation of $\tau$ in this form witnesses the $BCP_0^+$ condition if: $\exists \alpha < \Theta$ $\forall g \in \omega^{\omega}$ the downward $\subseteq$ closure of $\{A : \neg A\mathcal{R}\{g\} \text{ and } A \text{ has Wadge degree } < \alpha \}$ is $\{A : \neg A \mathcal{R} \{g\}\}$
To motivate Noah's question a bit for those who don't want to read between the lines: usually a relation on $\mathcal{P}(\omega^{\omega})$ is hard to get your hands around. Noah is interested in cases where you can recover $\tau$ completely just from the restriction of $\mathcal{R}$ to a nice, small (meaning, size $\leq 2^{\aleph_0}$) subclass of $\mathcal{P}(\omega^{\omega}) \times \mathcal{P}(\omega^{\omega})$. Then you can hope to represent $\mathcal{R}$ as a relation on $\omega^{\omega}$, and use all your tools from classical descriptive set theory.
Noah's first observation is that if a topological space has base of compact sets of size at most $2^{\aleph_0}$, there is a representation in which you can recover $\tau$ completely from the restriction of $\mathcal{R}$ to the finite powerset $\mathcal{P}_{Fin}(\omega^{\omega})$
His second observation is that under a certain condition, you can recover $\tau$ completely from the restriction of $\mathcal{R}$ to the $\Pi^0_1$ powerset $\mathcal{P}_{\Pi^0_1}(\omega^{\omega})$.
These conditions in $BCP_0^+$ and $BCP_0^-$ are conditions to ensure a (relatively) easy recovery process.
